For a real transcendental number ξ, let ωn∗(ξ) denote the supremum of all ω for which there exist infinitely many real algebraic numbers α of degree ≤n satisfying ∣ξ−α∣≤H(α)−ω−1, where H(α) is the naive height of the minimal polynomial of α. A celebrated result of Wirsing gives the uniform lower bound ωn∗(ξ)≥2n+1, which was improved significantly in a recent work of Poëls to 2−log2n. In this paper, we establish a p-adic counterpart of Poëls's result. Let p be a prime and ξ∈Qp be transcendental. Let ωn,p∗(ξ) be the supremum of all real numbers ω for which there exist infinitely many algebraic numbers α∈Qp of degree ≤n such that ∣ξ−α∣p≤H(α)−ω−1. We show that ωn,p∗(ξ)≥2−log2n−1. This improves the known lower bounds in the p-adic setting, namely the analogue of Wirsing's theorem, due to Morrison and Teulié.
arXiv:2410.09215v3·Updated Aug 18, 2026·Originally submitted Oct 11, 2024
Continued fractions have been generalized over the field of p-adic numbers, where it is still not known an analogue of the famous Lagrange's Theorem. In general, the periodicity of p-adic continued fractions is well studied and addressed as a hard problem. In this paper, we show a strong connection between periodic p--adic continued fractions and the convergence to real quadratic irrationals. In particular, in the first part we prove that the convergence in R is a necessary condition for the periodicity of the continued fractions of a quadratic irrational in Qp. Moreover, we leave several conjectures on the converse, supported by experimental computations. In the second part of the paper, we exploit these results to develop a probabilistic argument for the non-periodicity of Browkin's p-adic continued fractions. The probabilistic results are conditioned under the assumption of uniform distribution of the p-adic digits of a quadratic irrational, that holds for almost all p-adic numbers.
arXiv:2606.03472v2·Updated Aug 17, 2026·Originally submitted Jun 2, 2026
Let Q=−x12−x22−x32+x42 be the standard signature (1,3) quadratic form. To each non-degenerate rational plane L in the four-dimensional quadratic space (Q4,Q) we can naturally attach a periodic geodesic on the Bianchi orbifold SL2(Z[i])\H3 which records the position of L in the Grassmannian up to integer rotations. Moreover, each such plane L defines a CM point and a periodic geodesic on the modular curve through restriction of Q to L and its orthogonal complement. Lastly, the local isomorphism between SO1,3(R) and SL2(C) gives rise to a further periodic geodesic on the Bianchi orbifold. In this article, we exhibit a natural coupling of all the above objects and prove simultaneous equidistribution under a Linnik-type splitting condition. The main ingredient is the classification of joinings of higher-rank diagonalizable actions on homogeneous spaces due to Einsiedler and Lindenstrauss.
We prove that for every ergodic transformation T on an infinite standard probability space both the automorphism group (i.e. centralizer) Aut(T) and its reversing automorphism group are realizable as symmetry groups of graphings. This is an analogue of Sabidussi's realization of arbitrary graph-automorphism groups, and provides numerous examples of graphing automorphism groups carrying no compatible compact topology, answering a question of Lovasz'. Another consequence of discussion and ensuing constructions is the existence of large mutually locally-globally equivalent graphing families with highly variable symmetry.
arXiv:2406.02243v3·Updated Aug 17, 2026·Originally submitted Jun 4, 2024
A number m is called a de Polignac number(POL in short) if it can be expressed as the difference of infinitely many pairs of consecutive prime numbers. For any given r∈N, a set A is said to be Δr⋆ if for all sets S with ∣S∣=r such that A∩{s−t:s>t∈S}=∅. In this article we prove that the set POL is Δr⋆ with the specific computable value of r, where r=exp(O(50)). Then we prove that there exists a set E with ∣E∣≤250⋅∏i=149pi (where (pi)i is the enumeration of primes) such that N=⋃t∈Et−1POL.
Let K be a number field of degree D with maximal order OK. We show that under certain conditions on K, which in particular are always satisfied if D is odd or if D≥3 and K is primitive, the set of positive integers NK that can be expressed as a sum of two units in OK∗ is a finite effectively computable set. This result partially resolves an open problem posed by Tinková, Yatsyna, and the first author. We illustrate our method by explicitly computing NK for the smallest totally real quintic field K with Galois group S5.
We prove a lower bound for the Hausdorff dimension of weighted totally irrational singular vectors in affine-spanning common-base integral self-similar sets satisfying the open set condition. For the middle-third Cantor square, the bound improves the previously known explicit lower bound.
arXiv:2412.02278v3·Updated Aug 16, 2026·Originally submitted Dec 3, 2024
Tsukamoto (2022) introduced the notion of Bedford-McMullen carpet system, a subsystem of ([0,1]N×[0,1]N,shift) whose metric mean dimension and mean Hausdorff dimension does not coincide in general. The aim of this paper is to develop the mean dimension theory for Bedford-McMullen sponge system, which is a subsystem of (([0,1]r)N,shift) with arbitrary 3≤r∈N. In particular, we compute the metric mean dimension and mean Hausdorff dimension of such topological dynamical systems explicitly, extending the results by Tsukamoto. The metric mean dimension is a weighted combination of the standard topological entropy, whereas the mean Hausdorff dimension is expressed in terms of weighted topological entropy. We also exhibit a special situation for which the metric mean dimension and the mean Hausdorff dimension of a sponge system coincide.
arXiv:2608.11864v2·Updated Aug 15, 2026·Originally submitted Aug 12, 2026
For a positive integer n, let fn(X)=X4−nX3−6X2+nX+1 and let Kn=Q(ρn), where ρn is a root of fn. We determine all coincidences among these fields: for distinct positive integers m,n, Km=Kn⟺{m,n}∈{{1,103},{2,22},{4,956}}. Thus the three previously known equalities are the only ones. This extends Hoshi's finite-range classification to all positive integral parameters and, in particular, subsumes the uniqueness results of Pincus and Washington. The proof combines Hoshi's correspondence between equal simplest quartic fields and primitive solutions of a quartic Thue equation with estimates of Lettl--Pethő--Voutier for rational approximations to two of its real roots. A Gaussian-integer identity yields a lower bound for the denominator of the resulting rational approximation; the continued-fraction information and the approximation estimates then exclude every parameter exceeding 1000.
Federico Accossato, Nadir Murru, Giuliano Romeo, Giulia Salvatori
math.NT11J7011D8811J81
Given a field K, let K((T−1)) be the field of formal power series. Continued fractions in K((T−1)) can be defined by analogy with classical real continued fractions and have been widely studied. Some results establish the transcendence of elements of K((T−1)) arising from special families of continued fractions, but much remains to be explored. In this paper, assuming that K has characteristic zero, we improve the known analogues of the Maillet--Baker criteria for quasi-periodic continued fractions. A central tool that we prove is a quantitative version of Uchiyama's analogue of Roth's theorem in function fields, which gives an explicit bound for the number of exceptionally good rational approximations to an algebraic power series. This quantitative estimate also yields a Davenport--Roth-type upper bound on the growth of the denominators of the convergents of algebraic elements. Finally, we prove that palindromic continued fractions are either quadratic or transcendental, as in the real case, but using a different proof strategy.
The Shimura curve of discriminant D for D=14,15 is uniformized by a subgroup of an arithmetic quadrilateral Fuchsian group (2,2,2,q), where q=4,6. We relate the generator of the ring of quaternionic modular forms on this Shimura curve to explicit Heun functions for the quadrilateral group. We also discuss how the Picard-Fuchs equation of the associated family of abelian surfaces has solutions that are modular forms on XD(1)/WD, where WD is the full group of Atkin-Lehner involutions. This leads us to completely describe the rational exceptional sets of the associated Heun functions, and the algebraic values attained by the Heun function on these points, for example He(81,21;31,61,21,21;−112729)=(7522⋅33⋅53)61.
arXiv:2506.01736v2·Updated Aug 14, 2026·Originally submitted Jun 2, 2025
Motivated by the Berry-Tabor Conjecture and the seminal work of Rudnick-Sarnak, the fine-scale properties of sequences (anα)n∈Nmod1 with (an)n∈N⊆N and α irrational have been extensively studied in the last decades. In this article, we prove that for (an)n∈N arising from the set of rough numbers with explicit roughness parameters and any badly approximable α, (anα)n∈Nmod1 has Poissonian correlations of all orders, and consequently, Poissonian gaps. This is the first known explicit sequence (anα)n∈Nmod1 with these properties. Further, we show that this result is false for Lebesgue almost every α, thereby disproving a conjecture of Larcher and Stockinger [Math. Proc. Camb. Phil. Soc. 2020]. The method of proof makes use of an equidistribution result mod d in diophantine Bohr sets which might be of independent interest.
We develop an abstract Banach principle in vector lattices and apply it to obtain lattice-theoretic versions of the individual and maximal ergodic theorems, without recourse to any measure representation. Considering a sequence of bounded operators with values in a Dedekind σ-complete vector lattice endowed with a locally solid topology satisfying the σ-Lebesgue property, we prove that the set of points at which the sequence is order convergent is a closed subspace and coincides with the whole space whenever convergence holds on a dense subset. Investigating positive, power-bounded, mean ergodic operators on order continuous Banach lattices, we construct a topology on the universal completion induced by a strictly positive order continuous functional and prove that the Cesàro means converge in order in the universal completion and, in particular, uo-converge in the original lattice. Moreover, we introduce the notion of a superinvariant pair and derive a lattice-theoretic Hopf inequality together with a weak type estimate via band projections, which yields an abstract maximal ergodic theorem. A spectral-theoretic version of the theorem follows from classical Perron--Frobenius theory. Finally, we specialise the abstract framework to the model space L0(Ω) and revisit the classical Banach principle, the Hopf--Dunford--Schwartz theorem and Doob's martingale convergence theorem.
We investigate proper actions of arbitrary semigroups on separable locally compact metric spaces, where point orbits are allowed to escape to infinity. An action is pointwise uniquely ergodic when every compact orbit closure supports exactly one invariant probability measure and non-compact orbit closures support none. The associated ergodic map therefore assigns the selected probability measure to non-escaping points and the zero subprobability to escaping ones. Under the hypothesis that compact orbit closures admit at least one invariant measure, we establish that the weak* continuity of this ergodic map together with a vanishing at infinity condition is equivalent to the mean ergodicity of the Koopman representation on the space of continuous functions vanishing at infinity. In consequence, every such function and every finite signed measure split uniquely into invariant components and limits of coboundaries. The corresponding projections are obtained by integration against the ergodic map. Because this operator-theoretic characterization avoids explicit averaging schemes, it remains applicable even to semigroups without Følner sequences. When restricted to countable, discrete, bicancellative, and left amenable semigroups, these properties are shown to be equivalent to the uniform convergence of Følner averages and the weak-star continuity of their dual limits, extending classical results for group actions on compact spaces. Furthermore, we identify the space of ergodic measures with a compactified ergodic quotient, prove that the invariant measure simplex is Bauer, and show that these structural properties descend through proper factor maps. The theoretical framework is complemented by dynamical examples, including a continuously pointwise ergodic subshift that exhibits discontinuous entropy along the ergodic map.
We study mixing at a double exponential rate on analytic observables and ask how much of a map is remembered by its rate of mixing. For finite Blaschke products of the circle with a fixed point in the unit disk, and for the free semigroup actions they generate, we give a complete classification: the rate of mixing (no mixing, exponential, or double exponential) is determined by the multiplier of the generators at that fixed point, the invariant measure being the harmonic measure with a pole there. In the double exponential regime, the exponent equals logp, where p is the minimal local degree of the generators at the fixed point, and we show that this value is sharp. Consequently, the rate is not rigid: it is not stable under C1-perturbations and does not imply C1-conjugacy to affine models. Rigidity holds when the exponent is maximal for the degree: a map of degree q whose exponent attains logq is Möbius conjugate to an affine model.
arXiv:2607.19337v2·Updated Aug 13, 2026·Originally submitted Jul 21, 2026
Let pn denote the n-th prime. We prove that for every nonzero polynomial f(x)∈Z[x], there exist infinitely many positive integers n such that pn∤f(n).
arXiv:2604.17069v3·Updated Aug 13, 2026·Originally submitted Apr 18, 2026
Classically, Markov numbers are recovered as perfect matching numbers of domino snake graphs. We extend this correspondence by introducing weighted universal generalised snake graphs, or wug-snake graphs. These are weighted ordered bipartite graphs whose perfect matching sequences encode linear recurrences. To every wug-snake graph we associate a continuant matrix and prove that the determinant of this matrix equals the weighted perfect matching sum. We then introduce polyomino wug-tiles, bodies of wug-snake graphs that act linearly on state vectors. Every integer matrix admits a canonical polyomino wug-tile. Our main result identifies the wug-snake determinant of a tile representing a matrix A with the Markov-Davenport form of A. Consequently, algebraic and geometric Markov numbers of matrices and matrix semigroups can be expressed as weighted perfect matching determinants. Further we define Frobenius maps for matrix semigroups and discuss examples recovering classical Markov numbers and higher-dimensional lattice realisations.
In this paper, we study the random Laguerre tessellation, a weighted generalization of the Voronoi tessellation, generated by a general stationary marked point process. We first derive a nearly optimal sufficient condition on the generating marked point process which ensures that the resulting random Laguerre tessellation is well-defined, utilising the concept of tempered configurations to handle potentially unbounded weights. We then investigate how the three mixing properties - ergodicity, mixing and α-mixing - of the generating marked point process are preserved for the corresponding random Laguerre tessellation. Our approach combines standard approximation arguments with the properties of tempered configurations and the measurability of the Laguerre mapping.
arXiv:2608.02489v2·Updated Aug 13, 2026·Originally submitted Aug 3, 2026
In this paper, we establish quantitative Denjoy--Koksma type estimates for higher-dimensional quasi-periodic torus rotations. For Diophantine frequency vectors, we establish quantitative estimates on the discrepancy between Birkhoff averages and spatial averages for observables with various Besov-type regularities. By means of suitable Sobolev embeddings, these estimates yield, to the best of our knowledge, the sharpest currently available convergence rates for Hölder continuous observables. As applications, we obtain substantially improved quantitative homogenization results for Hamilton--Jacobi equations in spatially quasi-periodic settings, as well as nearly optimal statistical regularity estimates for invariant measures under perturbations.
arXiv:2607.18987v2·Updated Aug 13, 2026·Originally submitted Jul 21, 2026
We prove a refinement of a recent structural result in topological dynamics due to Glasscock, Koutsogiannis, Le, Moreira, Richter, and Robertson, who showed that in a minimal system, polynomial return-times differ by non-piecewise syndetic sets from those in its maximal infinite-step pronilfactor. In particular, we prove that the infinite-step pronilfactor can be replaced by the maximal k-step pronilfactor of the system, where k depends only on the given polynomials, with k−1 being equal to the Host-Kra complexity of the polynomials.
arXiv:2607.06907v2·Updated Aug 13, 2026·Originally submitted Jul 8, 2026
In this paper, we investigate weighted Birkhoff averages for toral translations associated with compactly supported weighting functions. By introducing several new analytical techniques, we establish optimal uniform convergence rates for almost all rotations and specific (or even all) initial points. Unlike the O(N−1) rate best achieved in classical ergodic theory, we show that these weighted averages exhibit polynomial or even exponential convergence. We establish the optimality of these convergence rates in multiple aspects, particularly concerning regularity indices across four distinct cases: finite differentiability, the C∞ class, logarithmic C∞ classes, and Gevrey classes. Our results demonstrate that the regularity of the observable essentially dictates the convergence rate; furthermore, we prove that in general settings, no alternative weighting function can yield a faster uniform rate. In contrast to the generically slow convergence of standard time averages, this work provides an optimal and nearly complete characterization of rapid convergence for weighted Birkhoff averages.
arXiv:2607.27774v2·Updated Aug 13, 2026·Originally submitted Jul 30, 2026
Finite logarithms can be defined in the ``poor man's adèle ring" A using Fermat quotients modulo sufficiently large primes. This ring contains Q and outside the trivial cases, Matsusaka and Seki have shown that finite logarithms cannot take non-zero rational values in A. Furthermore, a theorem of Silverman shows they are not zero, assuming the abc-conjecture. We extend these results to primes restricted to arithmetic progressions of the form p≡1modm by relating Fermat quotients to values of cyclotomic polynomials and their logarithmic derivatives. A signed version of the same argument shows unconditionally that the square of a finite logarithm cannot take non-zero rational values in A. As a further application, we show that, subject to the abc-conjecture, finite logarithms cannot be quadratic irrational elements of A in Rosen's theory of finite algebraic numbers.
arXiv:2208.08911v4·Updated Aug 12, 2026·Originally submitted Aug 18, 2022
In this paper, we study the quasi-stationary behavior of the one-dimensional diffusion process with a regular or exit boundary at 0 and an entrance boundary at ∞. By using the Doob's h-transform, we show that the conditional distribution of the process converges to its unique quasi-stationary distribution exponentially fast in the total variation norm, uniformly with respect to the initial distribution. Moreover, we also use the same method to show that the conditional distribution of the process converges exponentially fast in the ψ-norm to the unique quasi-stationary distribution. The rate of convergence of the conditional empirical measure to the quasi-ergodic distribution is also considered. Finally, two examples arising in population dynamics are also given to illustrate the main results.
We study quantitative edit-distance asymptotics for symbolic codings of irrational rotations x↦x+α on T in terms of the irrationality exponent μ(α), the supremum of μ∈R for which the inequality 0<∣α−p/q∣<q−μ has infinitely many solutions. For the binary coding determined by an interval [0,β), let WN be the set of length-N words arising from all initial points x under x↦x+α. We develop new techniques for estimating edit distance and compute the growth exponents of the edit-distance diameter diamE(WN). For every α∈/Q and almost every β∈(0,1), we show that (∗)N→∞limsuplogNlogdiamE(WN)=μ(α)μ(α)−1, and the corresponding liminf equals 1/2. When μ(α)−1 is at most the golden mean φ, the asymptotics (∗) hold for all β. However, for μ>1+φ, there is an uncountable set of α with μ(α)=μ for which the edit-distance exponents are strictly smaller than (∗) for uncountably many β. We also derive consequences for aperiodic circle homeomorphisms and Sturmian sequences. For rotations of Td coded by boxes, we prove that for almost every rotation vector, the common edit-distance exponent is d/(d+1). Finally, we raise the question of estimating edit-distance exponents for more general dynamical systems.
An analog of the classical Kronecker-Weyl theorem for weak uniform distribution is obtained for an arbitrary countably infinite independent subset of a discrete abelian group using a type of interpolation sets called epsilon-Kronecker sets. Specifically, the topological size of the set of homomorphisms inducing weakly uniformly distributed sequences is established by showing that it forms a dense G-delta set.
Given a non-atomic standard probability space (Ω,μ), we exhibit examples of non-amenable closed topological subgroups of Aut(Ω,μ) whose natural near-action on (Ω,μ) is whirly. This answers a 2010 question by Pestov in the negative. The argument proceeds via constructing Gaussian near-actions from topologically faithful unitary representations of non-amenable whirly Polish groups.
arXiv:2509.20596v2·Updated Aug 12, 2026·Originally submitted Sep 24, 2025
The use of data-driven control strategies on systems with not fully measurable states induces the problem of learning-based state observation. Motivated by this need, the present work proposes a data-driven approach for the synthesis of state observers for discrete-time nonlinear systems with measure-preserving dynamics. To this end, Kazantzis--Kravaris/Luenburger (KKL) observers are shown to be well-defined, where the observer design boils down to determining a nonlinear injective mapping of states and its pseudo-inverse. For its learning-based construction, the KKL observer is related to the Koopman operator, well-defined on the square-integrable function space and restrictable to a Sobolev-type reproducing kernel Hilbert space (RKHS). Hence, observer synthesis algorithms, based on kernel interpolation/regression routines for the desired injective mapping in the observer and its pseudo-inverse, are proposed in various settings of the available dataset -- (i) many orbits, (ii) single long orbit, and (iii) snapshots. Theoretical error analyses are provided, and numerical studies on a chaotic Lorenz system are demonstrated.
Let Ω(n) denote the number of prime factors of n, counted with multiplicities. We study the set AccΩ(x) of weak-∗ limits of the sequence N1∑n≤NδTΩ(n)x in σ-compact dynamical systems (X,T), demonstrating that if x∈X is quasi-generic for an ergodic measure μ, then μ∈AccΩ(x). This extends a result of Bergelson and Richter, who studied the problem in the setting of uniquely ergodic systems. We give a more precise description of the set AccΩ(x) in the case of the horocycle flow on non-compact quotients of SL(2,R). We show that for every non-periodic x∈X, in addition to Haar measure, there exists sequences (sn),(cn)⊆R such that 2π1∫−∞∞e−2r2νsn−2log∣1+cnr∣idr∈AccΩ(x), where {νsi}i≤k denotes the one parameter family of periodic measures in each of the k inequivalent cusps. Depending on Diophantine properties of the non-periodic point x, we show that AccΩ(x) contains a full two parameter family of such periodic measures, as well as the Dirac measure at each cusp. In particular, these results yield almost-everywhere divergence of pointwise averages along Ω(n) for the non-compact horocycle flow.
arXiv:2509.24982v2·Updated Aug 11, 2026·Originally submitted Sep 29, 2025
We characterize the bialgebraic varieties of the Γ function, that is, if V,W⊆Cn are irreducible affine algebraic variety which satisfy dimV=dimW and Γ(V)⊆W, then the equations defining V (and hence also W) either give an equality between coordinates, or set some coordinates to be constant. We also classify the case where V and W have the same dimension as varieties over the field of algebraic functions.
We study long-branched unimodal maps for which the sequence of values of the kneading map form a Sturmian word. We identify these kneading classes with those arising from the stunted Lorenz construction of Anušić, Bruin, and Činč, and their quadratic representatives with the maps constructed by Blé from rigid rotations. We compute the cutting and co-cutting times explicitly and establish criteria for the existence and nonexistence of absolutely continuous invariant probability measures. We also show that, for Lebesgue-almost every rotation angle, no finite-order S-unimodal realization satisfies the Collet-Eckmann condition.
arXiv:2606.03941v2·Updated Aug 10, 2026·Originally submitted Jun 2, 2026
The study on shift spaces in ergodic theory has extended beyond the classical setting, but there is room to discuss an extension of the Kolmogorov-Sinai entropy in the ergodic theoretical point of view. On the other hand, the rate distortion dimension recently attracted attention in mean dimension theory because it behaves like the Kolmogorov-Sinai entropy on dynamical systems in the ``large" spaces in which the usual entropy is in general infinite. According to these background, we investigate the connection between the Gibbs measure on the product spaces and the variational principle based on the rate distortion dimension. Concretely we calculate the rate distortion dimension of the Gibbs measure for a Hölder continuous potential depending on the first two coordinates and it satisfies the simplest case of themodynamic formalism based on the rate distortion dimension: the extension of the maximal measure of topological entropy. Notably, to the best of our knowledge, this is the first study to introduce Gibbs measures into mean dimension theory, and this connection allows the mean dimension with potential to be computed more naturally through the notion of the zero-temperature limit, in the spirit of thermodynamic formalism.
arXiv:2011.01182v3·Updated Aug 10, 2026·Originally submitted Nov 2, 2020
Motivated by Matui and Kerr's work on almost finiteness, we introduce a new finite approximation property for (possibly non-ample) étale groupoids called {almost elementariness}, which unifies and generalizes both almost finiteness and pure infiniteness. This property serves as a dynamical analogue of regularity properties of C∗-algebras. In support of this view, we prove that minimal almost elementary groupoids yield tracially Z-stable reduced groupoid C∗-algebras. Consequently, we obtain as a corollary that the reduced C∗-algebras of all minimal amenable second countable almost finite groupoids in Matui's sense are Z-stable and thus classifiable by the Elliott invariants. Two basic ingredients underlying the definition of almost elementariness are castles in groupoids and groupoid subequivalence, both of which are developed extensively in this work. Notably, building on our flexible framework of castles, we introduce the technique of nesting of castles, which play a key role in the proof of our main theorem. Besides our main theorem on tracial Z-stability, we also discuss in depth the relations between almost elementariness and other properties for groupoids such as effectiveness, the groupoid small boundary property, groupoid strict comparison and Matui's and Kerr's notions of almost finiteness.
Let (X,T) be a topological dynamical system and (M(X),T∗) be its induced system. For a non-empty compact subset K⊂X, we define M(K) as the set of Borel probability measures supported on K. In this paper, we systematically study the relationship between various entropies of (T,K) and of (T∗,M(K)). We show that: htopUC(T,K)>0⟺htopUC(T∗,M(K))=∞,htopP(T,K)>0⟺htopP(T∗,M(K))>0,htopB(T,K)>0⟹htopB(T∗,M(K))>0, where htopUC(T,K), htopP(T,K), and htopB(T,K) denote the upper capacity topological entropy, the packing topological entropy, and the Bowen topological entropy of K, respectively. Additionally, we present a counterexample involving a non-invariant set, demonstrating that the converse of the third assertion is not valid in general.
arXiv:2606.28771v2·Updated Aug 9, 2026·Originally submitted Jun 27, 2026
Xiaobo Hou, Wanshan Lin, Xueting Tian, Yi Yuan, Xutong Zhao
math.DS37A3537B4037D2037B10
This paper studies conditional entropy realization and weak* approximation by uniquely ergodic measures with compact support. We prove that, after fixing an admissible potential average and an entropy strictly below the entropy supremum over the corresponding average fiber, every invariant measure satisfying these two exact constraints can be approximated weakly* by uniquely ergodic measures with compact support satisfying the same constraints. Each approximating measure is the unique invariant measure on its minimal support, whose topological entropy equals the prescribed metric entropy. This result holds for three broad classes of systems: topologically expanding maps (including topologically Anosov systems), transitive countable Markov shifts, and symbolic systems with non-uniform structure. The proof uses a nested multi-horseshoe construction, with separate arguments addressing non-invertibility, non-compactness and non-uniformity.
Every regular N-gon defines a canonical family of regular polygons which are conforming to the bounds of the 'star polygons' determined by N. These star polygons are formed from truncated extended edges of the N-gon and the intersection points ('star' points) determine the parameters of the family. The First Family Theorem (FFT) shows how each star[k] point defines a matching S[k] 'tile' and the GeneralizedFFT shows how every S[k] can generate their own families with an S[k+1] on the left and (if space permits) an S[k'] on the right where for N even k' = N/2-k. If such a tile does exist S[k] and S[k'] will be called a 'dual' pair and share an edge. If N is even the S[1] tile and N will form a dual pair since N = S[N/2-1] relative to the S[k]. When the outer-billiards map Tau is introduced in Section 4, this shared edge will form an barrier to the dynamics so S[k] and S[k'] will be on opposite sides of two invariant regions and the shared edge can be extended to form a separatrix between regions which is similar to an 'integral' curve when N is regarded as a harmonic oscillator H. The Invariance Conjecture implies there will be EulerPhi(N)/2) distinct Ik to match the 'primitive' star[k] with gcd(k,N) = 1. These curves (also known as Lagrangian graphs) define what we call the eigen-states of N. The Twice-Odd Lemma says that N and N/2 will share eigen-states. The assumption that each S[k] can have right and left side families in W is the Reflection Principle which says that W must evolve both right and left handed. This principle is actually a property of N itself but the Dihedral group D(N) is Non-Abelian with regard to reflections and rotations. Therefore every regular N-gon is 'chiral' and -W and +W are distinct properties of N. Mapping them to the union cannot be smooth since the Jacobian J = 0, and the resonances should correspond to the S[K] dual pairs as solitons.
We prove a noncommutative topological factor theorem for irreducible lattices in products of real rank-one simple Lie groups. The intermediate C*-subalgebras between the reduced group C*-algebra and the boundary crossed product are exactly the crossed products arising from coordinate subproducts of the Furstenberg boundary. This follows from a more general theorem for product boundary actions, which also yields tree and mixed local-field versions. We finally show that the corresponding classification for the full flag action of SL3(Z) would imply ordinary ITAP.
It is well known that the computational realization of Gibbs probability measures, e−U(x)/Z, plays a central role in sampling and optimization. In this paper, we introduce two types of dynamics that exhibit rapid convergence towards these Gibbs measures. The mechanism driving this rapid convergence is the enhanced dissipation associated with these transport-diffusion dynamics. Motivated by these enhanced dynamics, we design numerical algorithms for sampling from the target Gibbs measure. Finally, we provide the corresponding particle systems that may yield other effective numerical samplers.
arXiv:2302.05240v5·Updated Aug 7, 2026·Originally submitted Feb 10, 2023
We show that if {φi}i∈Γ and {ψj}j∈Λ are self-affine iterated function systems on the plane that satisfy strong separation, domination and irreducibility, then for any associated self-affine measures μ and ν, the inequality dimH(μ∗ν)<min{2,dimHμ+dimHν} implies that there is algebraic resonance between the eigenvalues of the linear parts of φi and ψj. This extends to planar non-conformal setting the existing analogous results for self-conformal measures on the line.
arXiv:2607.25773v2·Updated Aug 7, 2026·Originally submitted Jul 28, 2026
Richard Canary, Tengren Zhang, Feng Zhu, Andrew Zimmer
math.DSmath.DGmath.MG37A1737A4022E40
We obtain orbital counting results for relatively Anosov groups with respect to linear functionals with finite critical exponent. Our counting results follow from an equidistribution result and rely crucially on previous equidistribution results obtained in our proof of counting results for periods. Our results generalize earlier work of Sambarino in the setting of Anosov groups.
This work builds on the breakthrough of OpenAI in finding the first nonsofic group. We analyze the underlying proof mechanism and find further applications. Let Γ<G be such that {g∈G:gΓg−1≤Γ} generates G as a group, and suppose that both Γ and G have property (T). If Γ is not normal, then the generalized wreath product (⨁G/ΓZ/2Z)⋊G is nonsofic. These hypotheses hold for explicit pairs of elementary groups over polynomial and Laurent polynomial rings, in which both groups are residually finite and Kazhdan.
Conditional expectation operators (CEOs) and their associated conditional mean embeddings (CMEs) play a central role across applied mathematics and machine learning, appearing in nonparametric regression, Bayesian inverse problems, and Koopman operator theory. A fundamental question is when a CEO maps a function space on Y into a prescribed function space on X, particularly a reproducing kernel Hilbert space (RKHS). We show that such mapping properties are characterized by the regularity of the Radon--Nikodym density of the conditional law, and establish a simple, verifiable sufficient condition under which the CEO is bounded and Hilbert--Schmidt. For RKHSs norm-equivalent to Sobolev spaces, this condition reduces to Sobolev regularity of the conditional density. The result yields a direct route to validate CME representations and error bounds for Galerkin-type and CME-based estimators. We verify the regularity condition in three settings: nonparametric regression, Bayesian inverse problems, and Koopman operator theory for stochastic dynamical systems. We show in each case that classical regularity results on the underlying probabilistic model imply the required mapping properties. The resulting framework offers a unified perspective on conditional expectation operators across probability, operator theory, kernel methods, and stochastic dynamics.
In this paper, we study the strong averaging principle for multiscale time-inhomogeneous stochastic systems driven by multiplicative α-stable processes with α∈(1,2). Based on Khasminskii's discretization approach, we first establish that the fast component processes with a frozen slow variable admits a periodic measure. We then prove the strong convergence of the slow subsystem to an averaged system that depends on the time scale ε. For any fixed ε, if the reciprocals of the two periods τ1 and ετ2 are rationally linearly independent, an important consequence is that the averaged system has random quasi-periodicity. Furthermore, by applying the ergodic theorem, we prove the strong convergence of the slow subsystem to another averaged system, a time-inhomogeneous SDEs independent of the time scale ε. Our result is also novel even in the time-homogeneous case for a fully coupled multiscale system with multiplicative α-stable noises. Finally, we apply the result to a climate-weather system.
arXiv:2606.17880v2·Updated Aug 6, 2026·Originally submitted Jun 16, 2026
We study the pair correlation statistics of orbits generated by maps on the interval. We show that under suitable mixing and multifractal assumptions, the pair correlation statistics of an orbit will almost surely exhibit the same asymptotic behaviour as a suitable sequence of i.i.d. random variables. We will also show that, under suitable hypotheses, the pair correlation statistics defined by two orbits will almost surely exhibit the same behaviour as two suitable sequences of i.i.d. random variables. Specific dynamical systems to which our results apply to include Gibbs-Markov maps and the Gauss map. We also give an example of a slowly mixing system for which the pair correlation statistics of an orbit almost surely behave distinctly to an i.i.d. sequence.
We adapt the theory of Ruelle-Pollicott resonances to annealed random dynamical systems generated by independent and identically distributed families of maps. Introducing annealed transfer and Koopman operators, we define resonances as elements of the point spectrum of the associated operators and establish a decorrelation formula relating these resonances to the asymptotic decay of annealed correlations. We then study several classes of systems for which the theory can be made explicit. First, we consider a family of piecewise expanding Markov maps of the interval, we construct Banach spaces adapted to the dynamics and obtain a complete description of the annealed resonance spectrum. Then we investigate an inverse spectral problem, proving realisability results for prescribed collections of complex numbers as resonances of suitably constructed annealed dynamical systems.
We prove that a Kazhdan group admitting a sofic embedding into a metric ultraproduct of symmetric groups with a centralizer that acts ergodically on the associated Loeb probability space is locally embeddable in finite groups (LEF). In particular, every finitely presented Kazhdan group admitting such an embedding is residually finite. The main technical theorem says that the centralizer of a sofic embedding of a Kazhdan group is itself a metric ultraproduct of permutation groups.
We prove a quantitative version of the convergence case of Khintchine's celebrated theorem in metric Diophantine approximation, but where the approximated points are restricted to lying on the parabola. A novel feature of our result is that unlike other results in literature, the approximating function is no longer required to be monotonic. This requires us to obtain explicit constants in classical number theoretic results, most notably in Burgess' bound for character sums in short intervals.
arXiv:2408.05963v4·Updated Aug 5, 2026·Originally submitted Aug 12, 2024
We establish non-asymptotic error bounds for the classical Maximal Likelihood Estimation of the transition matrix of a given Markov chain. Meanwhile, in the reversible case, we propose a new reversibility-preserving online Symmetric Counting Estimation of the transition matrix with non-asymptotic deviation bounds. Our analysis is based on a convergence study of certain Markov chains on the length-2 path spaces induced by the original Markov chain.
Benjamin Earp-Lynch, Simon Earp-Lynch, Omar Kihel, Pagdame Tiebekabe
math.NT11B3911D6111J86
We examine the equation ya=i=1∑kFni for positive integers y,a≥2 and k≥3. This equation can be expressed as a problem in terms of the Zeckendorf representations of integers. Using bounds on linear forms in logarithms and Baker-Davenport reduction methods, we are able to completely solve the equation for y≤25000 when k=3, for y≤1000 when k=4, for y≤40 when k=5, and for y≤3 when k=6.
arXiv:2303.12900v2·Updated Aug 4, 2026·Originally submitted Mar 22, 2023
We extend anti-classification results in ergodic theory to the collection of weakly mixing systems by proving that the isomorphism relation as well as the Kakutani equivalence relation of weakly mixing invertible measure-preserving transformations are not Borel sets. This shows in a precise way that classification of weakly mixing systems up to isomorphism or Kakutani equivalence is impossible in terms of computable invariants, even with a very inclusive understanding of ``computability''. We even obtain these anti-classification results for weakly mixing area-preserving smooth diffeomorphisms on compact surfaces admitting a non-trivial circle action as well as real-analytic diffeomorphisms on the 2-torus.
arXiv:2603.27258v2·Updated Aug 4, 2026·Originally submitted Mar 28, 2026
Bryna Kra, Joel Moreira, Florian K. Richter, Donald Robertson
math.DSmath.COmath.NT05D1011B1337A4411B30
We give a short proof of the fact that every set of natural numbers with positive upper Banach density contains the sum of two infinite sets. The approach simplifies earlier existing proofs.
We study the composition of temporal ergodic averaging with spatial averaging over shrinking metric balls, and determine sharp Orlicz endpoints for the corresponding unrestricted joint limit. For consecutive Birkhoff averages normalized by NΛq(N) (q≥0), the sharp Orlicz endpoint is Llogq+1L. In particular, the ordinary case q=0 yields an LlogL local joint convergence theorem under the Lebesgue differentiation property alone, while L1 fails even on the Euclidean interval, answering two questions of Young. The same Llogq+1L endpoint holds for prime averages for every q≥1. For arbitrary time sequences, LlogqL always suffices at the same normalization NΛq(N) (q≥1), and this endpoint is sharp in the Orlicz sense for fixed-base exponential sequences kn (k≥2) and for sequences with polynomial ratio separation, including n!. The positive results rest on a local stability principle: under the ball Lebesgue differentiation property, pointwise temporal convergence lifts to the local joint limit whenever the associated temporal maximal function admits an L1 majorant. The additional logarithm in the regular-time case comes from lifting restricted logarithmic maximal estimates from sets to general functions. The arbitrary-sequence theorem uses a dyadic decomposition instead. The lower bounds are local ∞-sweeping out constructions, obtained by weighted local sweeping out for polynomial-growth regular times, by residue constructions for polynomially ratio-separated sequences, and by digit constructions for fixed-base exponentials.
We study long-time Strichartz estimates for the one-dimensional Schrödinger equation with quasi-periodic initial data. For two-frequency data with an algebraic frequency ratio, we observe that the behavior of the linear Schrödinger evolution changes depending on the algebraic degree of the ratio. Making use of this observation, we improve the Strichartz estimates on long time intervals. We also prove an endpoint L4 Strichartz estimate. Our proofs use Roth-type Diophantine inequalities and Vinogradov-type mean value estimates for the Parsell--Vinogradov systems.
We study sets of (measurable) returns in countable groups G, namely sets of the form {g∈G:μ(A∩TgA)>0} arising from measure-preserving actions. Extending a result of Bergelson, we show that sets of returns in G×G contain subsets of the form B×B, where B is large with respect to suitable notions of largeness that remain meaningful even for non-amenable groups. As a consequence, if G is amenable, then every sufficiently large subset A⊆G×G satisfies B×B⊆AA−1 for some large set B⊆G. We also investigate when sets of returns in G contain product sets BB with B large. In contrast with the Cartesian-product phenomenon above, this problem is considerably subtler in non-abelian groups and is closely connected to `symmetric correlation functions', namely functions of the form g↦μ(Tg−1A∩TgA). We use this connection to show that, for broad classes of amenable groups - including finitely generated nilpotent groups and certain solvable non-nilpotent groups, every sufficiently large set A⊆G contains a large subset B satisfying BB⊆AA−1. Finally, we establish polynomial analogues of these results for finitely generated nilpotent groups, extending earlier work of Bergelson and Ruzsa.
We establish exponential mixing for the randomly forced and weakly damped KdV equation in L2(T). The noise is bounded, localized, and degenerate in high frequencies. Our proof relies on a general probabilistic framework in [11,33], nonlinear smoothing for KdV and its linearization via normal form transformation, and stabilization of the system by localized force. This paper continues a series of works connecting asymptotic compactness, control theory, and ergodicity and mixing for randomly forced dispersive PDEs.
Many mixing dynamical systems (X,T,μ) are known to satisfy the hitting time statistics result r→0limμ{x:τB(y,r)(x)>t/μ(B(y,r))}=e−t, for μ-almost every y, where τB(y,r)(x) is the first hitting time of x to the ball B(y,r). Taking a different point of view, we fix x and consider τB(y,r)(x) as a function of y. We call this the visiting time of y from x, i.e. the time it takes for y to get a visit from x within a neighbourhood of radius r. We prove that r→0limμ{y:τB(y,r)(x)>t/μ(B(y,r))}=e−t, for μ-almost every x. As a byproduct we obtain a new method of proof for hitting time statistics.
We prove a nine-distance theorem for Kronecker sequences on flat three-tori. That is, we show that among the first N orbit points, at most nine distinct positive nearest-neighbour distances occur. This proves the conjecture of Haynes and Marklof. An example of Dettmann shows that nine is optimal. More generally, we prove that on a flat d-dimensional torus the number of such distances is at most 2d+1. The main tool is a new growth theorem for the denominators q1<q2<⋯ of best simultaneous approximations in a d-dimensional inner-product space, which is of independent interest. We prove that, whenever qn+2d is defined, either qn+2d≥2qn+1, or the indices 1,…,2d can be partitioned into disjoint pairs {j,k}, j<k, such that qn+k=qn+qn+j. In particular, qn+2d≥min{2qn+1,qn+qn+2d−1}≥qn+qn+1.
arXiv:2605.30606v2·Updated Aug 1, 2026·Originally submitted May 28, 2026
In 2007, Adamczewski and Bugeaud introduced the notion of the Diophantine exponent of an infinite word as a quantitative measure of repetition, leading to new transcendence results for real numbers whose expansions in an integer base are sufficiently simple. In the present article, we introduce the refined Diophantine exponent, which detects weaker forms of repetition while preserving the full strength of the classical approach. This new exponent applies in situations where repetition is partially obscured by some form of noise. Related ideas already appear in the work of Corvaja and Zannier in 2002 and, more recently, in the works of Kebis, Luca, Ouaknine, Scoones, and Worrell. Our approach provides a unified framework that recovers and extends these results, as well as those of Adamczewski and Bugeaud. We also develop quantitative refinements of this method, leading to results about transcendence measures. The recent breakthrough of Bell, Diller, and Jonsson in the context of algebraic dynamics is partly based on a similar idea, which also served as a motivation for the present work.
We study the generating series associated with the degree sequence of a monomial self-map of a projective toric variety. We establish conditions under which this series has its circle of convergence as a natural boundary, and hence is a transcendental, non-holonomic function. In the case of toric surfaces, our results are sharp; moreover, we answer a question of Bell by proving that its reduction modulo p is transcendental for all but finitely many prime numbers p.
arXiv:2607.27932v2·Updated Aug 1, 2026·Originally submitted Jul 30, 2026
The present work investigates the asymptotic behaviours at the zero-noise limit of the first near collision-time and first near collision-location between a pair of independent d-dimensional Brownian-driven self-stabilizing (McKean-Vlasov type) diffusions. These asymptotic are considered in a peculiar setting where the systems evolve in a bi-stable landscape and collisions are only triggered by the combined action of the Brownian noises. As the Brownian perturbations fade away, we show that the near collision-time increases at an explicit exponential rate and that related collision-locations persist in specific regions of the space. These results are mainly derived by tailoring classical Freidlin-Wentzell's exit-time (and exit-location) estimates into collision estimates. Similar asymptotic are established for related mean-field interacting particle system approximation, and for the one-dimensional case (where true collisions can be examined
For a set V of n-1 distinct positive integers write LR(V) = max_t min_{v in V} ||v t||, where ||x|| is the distance from x to the nearest integer; V is tight if LR(V) = 1/n, the value predicted by the Lonely Runner Conjecture. Goddyn and Wong (Integers 6 (2006), #A38) classified the tight sets obtained from the baseline [n-1] by replacing one speed r with a multiple mr, and proved that for a fixed r only finitely many non-multiple replacements can be tight, remarking that this "partially explains" why the two sporadic tight sets {1,3,4,7} and {1,3,4,5,9}, in which the speed 2 is replaced by an odd number, have no analogues. We make their finiteness effective and settle the case they singled out. Let U(n,r) be the region left uncovered when speed r is deleted from the baseline. We compute the length of every connected component of U(n,r) exactly, in both regimes 2r > n-1 and 2r <= n-1, in terms of an arithmetic quantity I(n,r). Since a connected set on which the inserted speed w must stay 1/n-close to the integers cannot be longer than 2/(wn), this yields the explicit necessary bound w <= 4rI/(2s-I) with s = n-r, and hence: if ([n-1] minus {r}) union {w} is tight with 2r <= n-1, then n <= 6r. This improves the constant implicit in Goddyn and Wong's finiteness theorem from 12 to 6 and makes the classification of single-speed modifications a finite computation for each n. Carrying the computation out for r = 2 we obtain the complete classification: ([n-1] minus {2}) union {w} with w > n-1 is tight if and only if (n,w) = (5,7) or (6,9); the same method disposes of r = 3 entirely. We also report exhaustive censuses in exact rational arithmetic, and note that the natural guess that tight sets have all speeds below 2n is false, a counterexample being the Goddyn-Wong set {1,...,29,31,90} with n = 32.
Matheus G. C. Cunha, Douglas D. Novaes, Gabriel Ponce
math.DS37E9937A9926A1828A7828A8034A3637C29
In this paper, we study a class of one-dimensional piecewise maps defined by infinitely many smooth expanding branches. This class arises naturally in the context of non-smooth dynamical systems and includes the first-return maps locally defined near sliding Shilnikov connections. By means of the theory of conformal iterated function systems (CIFS), we investigate several dynamical properties of these maps as well as the topological complexity of their invariant sets. In particular, we show that the dynamics restricted to the invariant set is topologically conjugate to the shift on NN. We also establish the existence of a unique conformal measure that is invariant and ergodic under the map.
For stochastic differential equations, we establish a relationship between the measure-theoretic entropy of the stochastic flow and the rate of volume growth of stable submanifolds under iteration. By combining the result of Kifer and Yomdin (1988), we show that under a suitable integrability condition, for systems with C∞ coefficients, the measure-theoretic entropy is upper semicontinuous with respect to the coefficients of SDEs.
arXiv:2312.06860v2·Updated Jul 31, 2026·Originally submitted Dec 11, 2023
This article deals with harmonic analysis on affine buildings. Its main goal is to construct suitable kernels associated to a discrete multitemporal wave equations on the latter spaces, the long-standing motivation being to contribute to progress in arithmetic quantum unique ergodicity (AQUE) on certain Riemannian manifolds.
We investigate manifestations of the Nilpotent Heuristic, which posits that recurrence and convergence phenomena known for measure-preserving ZD-systems extend to nilpotent group actions. Our main results establish seminorm estimates and limiting formulas for multiple ergodic averages arising from actions of 2-step nilpotent groups. In particular, if T1,…,Tℓ are totally ergodic and generate a 2-step nilpotent group, then N→∞limN1n=1∑NT1nf1⋯Tℓnℓfℓ=j=1∏ℓ∫fjdμ in the L2 norm for all bounded functions f1,…,fℓ; the same holds for any distinct-degree polynomial iterates. We also obtain popular-common-difference versions of the polynomial Szemeédi theorem in the same setting. In a different direction, our approach allows us to completely resolve the joint ergodicity conjecture for multidimensional polynomials and ZD-systems; we also present an example showing that, surprisingly enough, the 2-step nilpotent analog fails. We conclude with many open problems concerning joint ergodicity, seminorm estimates, and the structure theory of nilpotent systems.
For the quadratic family fc(z)=z2+c, the only parameters in the Mandelbrot set M for which the Julia set Jc has Hausdorff dimension 1 are c=0 and c=−2. Near c=0, Ruelle's theory gives a real-analytic expansion of the dimension. The tip c=−2 of M, however, is a non-hyperbolic parameter and the dimension function c↦dimH(Jc) is highly discontinuous there. We prove the sharp first-order asymptotic for the lower envelope of the Hausdorff dimension at the tip: If c∈M then dimH(Jc) lies asymptotically above 1+Ω∣c+2∣ with the Jaksztas constant Ω=32πlog21. This is a surprisingly precise contribution to the Yoccoz problem about unfolding attractors. The proof develops a thermodynamic formalism for degenerating families of box mappings. At each scale, for parameters c→−2, the induced dynamics exhibit a uniform property of exponential tails, generating improved control of their pressure functions.
By extending the Rokhlin-Sinai machinery relating to the entropy and countable Lebesgue component in the spectrum, we establish that the dynamical Möbius-Sarnak process has a countable Lebesgue component. Inspired by recent work of M. Lin and the author, we extend the notion of spectral measure to all operators on Banach spaces. This generalization is further motivated by the Bellow-Losert extension of Wiener's notion of the spectral measure of sequences. Furthermore, we establish unconditionally that the topological entropy of the Möbius flow is given by π26log3. Among other consequences, we recover a recent result by el Abdalaoui-Nerurkar which asserts that for any quasi-generic measure for the Möbius function, the Möbius flow equipped with this measure has a countable Lebesgue component in its spectrum. It follows that the Sarnak Möbius orthogonality conjecture holds for any topological dynamical system with singular spectrum. We further show that all the potential spectral measures of he Möbius function are absolutely continuous with respect to Lebesgue measure.
arXiv:2006.07646v2·Updated Jul 30, 2026·Originally submitted Jun 13, 2020
It is shown that Sarnak's Möbius orthogonality conjecture is fulfilled for the compact metric dynamical systems for which every invariant measure has singular spectra. This is accomplished by first establishing a special case of Chowla conjecture which gives a correlation between the Möbius function and its square. Then a computation of W. Veech, followed by an argument using the notion of `affinity between measures', (or the so-called `Hellinger method'), completes the proof. We further present an unpublished theorem of Veech which is closely related to our main result. This theorem asserts, if for any probability measure in the closure of the Cesaro averages of the Dirac measure on the shift of the Möbius function, the first projection is in the orthocomplement of its Pinsker algebra then Sarnak Möbius disjointness conjecture holds. Among other consequences, we obtain a simple proof of Matom{ä}ki-Radziwiłl-Tao's theorem and Matom{ä}ki-Radziwiłl's theorem on the correlations of order two of the Liouville function.
We study the almost Golomb equation of order r, a(S(n))=n, where S(n)=a(n)+a(n−1)+⋯+a(n−r+1), for nondecreasing sequences of positive integers. Its greedy solution is r-regular in the sense of Allouche and Shallit. Beyond that one, and for every non-square order r, the equation has another solution, an inhomogeneous Beatty sequence a(n)=⌊n/r+d⌋ where d is a parameter. No other positive slope occurs, and the equation holds for all n≥r exactly when d lies in an explicit closed interval depending on r. That interval is a single point when r=2 and has positive length for every non-square r≥3. Iterating the equation gives, for k≥1, a∘k(S(n))=a∘(k−1)(n), with a∘0 the identity. Its Beatty solutions of positive slope are again of the form ⌊n/r+d⌋, and the sets of admissible d form an increasing chain. The equation for k holds exactly when a∘(k−1) agrees at n and at a(S(n)). We determine these solutions for k=2 when r=2 and r=3. At the right endpoint of the r=2 interval the first equation fails on a thin set of indices, which we identify as the return times of an irrational rotation to an explicit interval. The proofs combine equidistribution with an exact computation in Z[r] over a finite range of orders.
arXiv:2508.17272v2·Updated Jul 30, 2026·Originally submitted Aug 24, 2025
In 1971, C. Fefferman established a higher dimensional extension of the celebrated Carleson--Hunt theorem which gives pointwise almost everywhere convergence of the partial Fourier sums of functions in Lp(T),1<p<∞. More precisely, Fefferman proved a maximal function bound for polygonal Fourier partial sums of functions in Lp(Td),p>1. In this note, we extend Fefferman's maximal function bound to strong r-variation norm bounds whenever r>2 as well as uniform 2-oscillation bounds. Furthermore, for functions in L2(Td), we establish r-variational and 2-oscillation bounds for partial Fourier sums over nested rectangles whenever r>2.
Frederik Broucke, Máté Matolcsi, Szilárd Gy. Révész
math.NT11J1343A35
In a blogpost in 2009, Gowers raised a possible approach to Littlewood's conjecture in Diophantine approximation, leading to a question about the existence of sufficiently many points in the unit cube such that "hyperbolic distance" between any two of them is large. In this note we answer this question by an explicit construction. This shows that this approach to prove Littlewood's conjecture cannot work, unless some further refinements are added.
This paper concerns the study of the existence of a unique equilibrium state for a Hölder continuous function under the dynamics of a holomorphic correspondence defined on the Riemann sphere. We mainly work with the correspondence restricted on the support of the Dinh-Sibony measure and identify topologically interesting correspondences, namely distance expanding ones. Further, we consider Hölder continuous potentials defined on the support of the Dinh-Sibony measure, for which we prove the uniqueness of equilibrium state. Along the way, we also prove some interesting topological results related to holomorphic correspondences. Finally, we establish a result connecting the Ruelle operator for holomorphic correspondences and the unique equilibrium state under a suitable hypothesis. The concluding part of the paper is devoted to some discussion related to the hypothesis involved and providing some examples.
Let F2 be the finite field with two elements and let F2ω denote the countably infinite-dimensional vector space over F2. We show that, unlike the case of polynomial maps p:Z→Z vanishing at zero which are always good for nice recurrence, there are polynomials p:F2ω→F2ω with p(0F2ω)=0F2ω which fail to have this property. This disproves a conjecture of Bergelson and McCutcheon (c. 2000), which predicted that for any countably infinite abelian groups H and G, every polynomial map p:H→G with p(0H)=0G is good for nice recurrence. Moreover, we develop a dynamical mechanism which shows that the magnitude of intersections along polynomial paths is degree-sensitive (even when one considers only weakly mixing systems). Among other things, we also show that the Furstenberg-Sarkozy theorem for F2ω-valued polynomials of degree at most d vanishing at zero is equivalent to a d-dimensional symmetric-difference weakening of the density polynomial Hales-Jewett conjecture. Thus, as we explain in detail in this paper, our observations not only shed new light on the phenomenon of polynomial recurrence but also constrain possible strategies for proving or disproving the density polynomial Hales-Jewett conjecture.
The classification of perfect e-codes over an arbitrary alphabet of size q is complete for e>2. In the case of non prime power q, it is conjectured that no perfect 2-codes exist. We confirm this conjecture in a number of situations, including the case where q=2αpβ with p prime, α and β positive integers, and either α≤20, or α sufficiently large.
arXiv:2601.03999v3·Updated Jul 28, 2026·Originally submitted Jan 7, 2026
We prove quantitative polynomial Wiener--Wintner theorems in a very general setup, including measure-preserving actions of nilpotent Lie groups. Our results apply both to ergodic averages and to averages with singular integral weights. The proof relies on the generalized polynomial Carleson theorem developed in the companion paper by van Doorn, Srivastava, and the authors.
Let P(Y1,...,Yd) be a certain fixed homogeneous polynomial of integral coefficients. In this paper, we establish a quantitative equidistribution criterion for the ergodic averages along Ω(∣P(n1,...,nd)∣). Consequently, by an estimate of Lachand, we prove the following variant of a theorem of Bergelson and Richter: if P is an irreducible binary cubic form and (X,T) is a uniquely ergodic system with unique invariant measure μ, then for any x∈X and f∈C(X), limN→∞N21∑∑n1,n2⩽Nf(TΩ(∣P(n1,n2)∣)x)=∫Xfdμ. Moreover, we prove in the appendix a related conjecture of Céspedes and Donoso over number fields.
A set of m distinct positive integers {a1,…am} is called a D(q)-m-tuple for nonzero integer q if the product of any two increased by q, aiaj+q, i=j is a perfect square. Due to certain properties of the sequence, there are many D(q)-Diophantine triples related to the Fibonacci numbers. A result of Baćić and Filipin characterizes the solutions of Pellian equations that correspond to D(4)-Diophantine triples of a certain form. We generalize this result in order to characterize the solutions of Pellian equations that correspond to D(l2)-Diophantine triples satisfying particular divisibility conditions. % Subsequently, we employ this result and bounds on linear forms in logarithms of algebraic numbers in order to classify all D(9) and D(64)-Diophantine triples of the form {F2n+8,9F2n+4,Fk} and {F2n+12,16F2n+6,Fk}, where Fi denotes the ith Fibonacci number.
Nikos Frantzikinakis, Oleksiy Klurman, Joel Moreira
math.COmath.NT05D1011N3711B3037A44
We address partition regularity problems for homogeneous quadratic equations. A consequence of our main results is that, under natural conditions on the coefficients a,b,c, for any finite coloring of the positive integers, there exists a solution to ax2+by2=cz2 where x and y have the same color (and similar results for x,z and y,z). For certain choices of (a,b,c), our result is conditional on an Elliott-type conjecture. Our proofs build on and extend previous arguments of the authors dealing with the Pythagorean equation. We make use of new uniformity properties of aperiodic multiplicative functions and concentration estimates for multiplicative functions along arbitrary binary quadratic forms.
Nikos Frantzikinakis, Oleksiy Klurman, Joel Moreira
math.COmath.NT05D1011N3711B3037A44
We address a core partition regularity problem in Ramsey theory by proving that every finite coloring of the positive integers contains monochromatic Pythagorean pairs, i.e., x,y∈N such that x2±y2=z2 for some z∈N. We also show that partitions generated by level sets of multiplicative functions taking finitely many values always contain Pythagorean triples. Our proofs combine known Gowers uniformity properties of aperiodic multiplicative functions with a novel and rather flexible approach based on concentration estimates of multiplicative functions.
Liouville numbers form a classical class of transcendental real numbers characterized by exceptionally strong rational approximations. A theorem of Maillet shows that non-constant rational functions with rational coefficients preserve the Liouville property, motivating a question of Mahler on whether analogous phenomena hold for transcendental functions. In this paper, we address this problem for real functions of finite smoothness. For any ε>0, we construct an uncountable set of Ck-functions on R, dense with respect to the topology of uniform convergence on compact sets, mapping Q into itself and satisfying den(f(p/q))≤q2k+ε, and deduce that such functions preserve Liouville numbers. In contrast, we prove a rigidity result about a C2k+1-function mapping Q into itself and satisfying den(f(p/q))≪qk.
Let w=(w1,w2) satisfy w1>w2>0 and w1+w2=1. For every function f:[0,∞)→(0,∞) with f(t)→0, we prove that the set of w-singular vectors whose weighted shortest-vector function satisfies Wx(t)≥logf(t) for all sufficiently large t has Hausdorff dimension sw=2−(1+w1)−1, equal to the full Hausdorff dimension of Singw(2). For every 0<ν<1 and μ>0, a separate power schedule gives Hausdorff dimension sw for the weighted uniform approximation set with rate Q−1exp(−μ(logQ)ν), whereas the lower-envelope theorem gives the same dimension for its complement in Singw(2). We give a direct proof of the lower-envelope theorem by adapting the self-affine construction of Liao--Shi--Solan--Tamam to a variable sequence of return times and using an empty-denominator-window argument to control intermediate cusp excursions without further pruning the tree.
We study quenched mixing rates for random compositions of two classes of interval maps with two indifferent fixed points and a singularity at the origin: Pikovsky maps and Grossmann--Horner maps. For the Pikovsky family, each fibre map preserves Lebesgue measure, so the equivariant sample measures are given by μω=m. For the Grossmann--Horner family, we construct an equivariant family (μω)ω∈Ω of absolutely continuous probability measures. Using random Young towers, we prove quenched polynomial decay of both future and past fibre correlations for bounded observables against Hölder observables. The rates are determined by quenched return time tail estimates obtained from endpoint drift bounds for the random cocycle.
Nearly linear recurrences generalise linear recurrences and can be represented as special cases of both linear time-invariant systems in control theory and linear-constraint loops in program analysis. We formulate the Positivity Problem for such recurrences: given a recurrence and initial values, decide whether every sequence satisfying the recurrence is termwise nonnegative. This problem generalises Positivity for linear recurrence sequences and is a special case of halfspace non-reachability for linear time-invariant systems. Our main result is a decision procedure for order-2 recurrences. The termination of the procedure relies on a transcendence theorem of independent interest: we prove that certain convergent series obtained by summing the absolute values of terms of algebraic linear recurrence sequences are transcendental.
For c>1 and an integer radix b≥2, we study the positive integers m for which mbk≤cm<(m+1)bk for some k≥0; for integer c, this is the self-prefix leading-digit condition. We derive an exact shrinking-target criterion; for c≥2, an exact signed-discrepancy identity isolates both infinitude and the conjectural logarithmic count. For c≥2 with nonintegral logarithmic slope, Lambert W−1 inversion produces a candidate sequence with an eventual two-gap law and an exact counting formula; for (c,b)=(2,10) all consecutive candidate gaps are 3 or 4. For algebraic c with irrational logbc, the Lambert-root phases satisfy deterministic moving-target asymptotics in an explicit nontrivial power range strictly below the critical scale. For irrational logarithmic slope, actual hits obey fixed-difference and arithmetic-chain rigidity; for multiplicatively independent integer parameters, coherent endpoint hits at floor resonance centers force every intermediate term. Finally, set ρ={logbc}. For fixed multiplicatively independent integers c,b, an interpolated continued-fraction locator has bit complexity O(N1−1/νpolylogN) for every ν>μ(ρ). We give an explicit certified instance for (2,10), whose infinitude remains open.
arXiv:1806.10884v3·Updated Jul 26, 2026·Originally submitted Jun 28, 2018
We consider rank one measure preserving transformations g and the corresponding Koopman unitary operators U(g). It is known that a generic (in the sense of Baire category) measure preserving transformation has rank one, spectral measure of U(g) is purely singular and is given by a Riesz product. For such transformations we write explicitly spectral decompositions of operators U(g) in generalized eigenfunctions.
In this paper, we show that for any S-unimodal map T on [0,1] with a non-flat critical point the Lyapunov exponent exists for Lebesgue almost every point and is equal to a constant λT∈R. Moreover, λT=0 if and only if T admits neither an absolutely continuous T-invariant probability measure with positive entropy nor a strictly stable periodic orbit. Consequently, if an S-unimodal map with a non-flat critical point is infinitely renormalizable or non-statistical then for Lebesgue almost every x∈[0,1] the Lyapunov exponent along the orbit of x exists and is equal to 0. A key ingredient is the following result of independent interest. If an S-unimodal map with a non-flat critical point has no periodic attractor then for Lebesgue almost every x∈[0,1] the lower Lyapunov exponent along the orbit of x is non-negative. This shows that, in the absence of periodic attractors, exponential contraction cannot occur along the orbit of Lebesgue almost every point.
arXiv:2505.02189v2·Updated Jul 25, 2026·Originally submitted May 4, 2025
We study hyperbolic components, also known as tongues, in the Double Standard Map family comprising circle maps of the form: fa,b(x)=(2x+a+πbsin(2πx))mod1,a∈R/Z,0≤b≤1. We prove simple connectedness of tongues by providing a dynamically natural real-analytic uniformization for each tongue. For maps in a tongue, we characterize the unique maximal subset of the circle on which fa,b is Devaney chaotic. We also show that the Hausdorff dimension of this maximal chaotic set varies real-analytically inside a tongue.
A set A⊆N is called complete if every sufficiently large integer can be written as a sum of distinct elements of A. It is strongly complete if it remains complete after one deletes finitely many elements from it. We show that A⊆N is strongly complete whenever A∩(2k,2k+1]≥6 for every sufficiently large k∈N, and a∈A∑∥aθ∥=∞,∀θ∈R∖Z. In particular, this resolves a 1961 conjecture of Erdős. The proof builds on previous work of Bergelson and Simmons. In fact, our approach allows us to establish a more general strong-completeness criterion with suitable ordered blocks in place of dyadic intervals. We also discuss some applications of our results as well as their connections to a few other interesting problems, including two completeness problems of Erdős and Graham.
arXiv:2603.19717v3·Updated Jul 24, 2026·Originally submitted Mar 20, 2026
Indistinguishability is a form of ergodicity, introduced by Lyons and Schramm for percolation clusters, that has become a fundamental notion in the theory of random infinite graphs. We prove the indistinguishability of connected components for a broad family of random one-ended or two-ended oriented forests on unimodular graphs using a new approach. We unify these models by introducing `coalescing Markov trajectories' (CMTs), which encompass a wide range of classical coalescing models, including river models and coalescing random walks. We also establish the indistinguishability of level-sets, a problem that has not previously been studied and lies beyond the scope of existing techniques. Using the latter, we prove that the clusters of the stationary voter model on a unimodular graph are indistinguishable. The core of the proof approach is the reduction of the indistinguishability of the components (respectively, level-sets) to the ergodicity (respectively, tail triviality) of the ancestry chain of the root. This shows a structural property that the ancestry chain is the fundamental object governing indistinguishability. For CMTs, the proof is completed by proving the tail triviality of Markov chains on unimodular random graphs, which is of independent interest. The flexibility of the approach is illustrated by further applications: It yields indistinguishability results for some point-map models on Bernoulli and Poisson point processes, including Howard's model and the strip point-map. It also yields a new and substantially simpler proof of indistinguishability for the wired uniform spanning forest, which is the only one-ended model for which indistinguishability was previously studied in the literature.
Let K be a number field with ring of integers OK, and let B be an Erdős family of ideals in OK. We prove that the associated B-free subshift (XB,(Sa)a∈OK) is intrinsically ergodic: it carries a unique measure of maximal entropy, which we identify explicitly as a relatively independent extension of the Haar rotation on ∏b∈BOK/b. This is the first proof of intrinsic ergodicity for B-free systems beyond dimension one, and relies on the work of Araújo--Dymek--Kułaga-Przymus. Via their reductions, we also settle the k-free and B-free lattice-point cases and the k-free number-field case. We give two independent proofs of the underlying rigidity statement: one through a single-site relative-entropy argument, and one through an exact-tiling realisation of Peckner's induce-and-split scheme.
Since Krengel's work [Kre78] in 1978, it has been widely known that no effective rate of convergence exists in the ergodic theorem. For toral translations, however, by choosing appropriate weights one can accelerate the convergence of ergodic averages to an exponential rate, but this intuitively requires both highly nonresonant frequencies and very regular observables. In this paper, we uncover a new phenomenon: even for any given nonresonant frequency, there exists a non-trivial family of weights and observables of low regularity such that the weighted Birkhoff averages along quasi-periodic orbits converge at a quantitative, uniform, and exponential rate. This not only yields a finer understanding of the deep interaction between nonresonance and regularity in ergodic theory, but also stands as a weighted counterpart to a Yoccoz-type result [Yoc80,Yoc95].
arXiv:2603.22045v3·Updated Jul 23, 2026·Originally submitted Mar 23, 2026
Let b≥2 be an integer. Using Sturmian words we describe all irrational real numbers ξ such that the image in R/Z of the sequence (ξ(−b)n)n≥0 is contained in an interval of length b−1+b−2−b−3. In previous work (arXiv:2603.16794) we showed that the image cannot be contained in a shorter interval.
arXiv:2512.09528v2·Updated Jul 23, 2026·Originally submitted Dec 10, 2025
We study the hyperbolic entropies of foliations obtained by suspensions of a representation, in the sense of Dinh, Nguyên and Sibony (topological and measure-theoretic). We establish a link between this type of entropy and an adapted version of an entropy defined by Ghys, Langevin and Walczak for pseudo-groups of homeomorphisms. Such a link has various consequences. Among them, it implies that the hyperbolic entropy of foliations is not invariant by diffeomorphisms, and that a minimal entropy suspension admits an invariant measure. Finally, this allows us to study thoroughly the simple case in which the image of the representation is isomorphic to~Z. In that case, we give the first exact estimate of the hyperbolic entropy, and prove a Brin--Katok type theorem and a variational principle, relying strongly on the standard ones for the entropy of maps.
Daeyeoul Kim, Zekiye Pinar Cihan, Zeynep Demirkol Ozkaya, Ilker Inam
math.NT11D4511J86
In the present paper, we identify all Jacobsthal numbers that may be expressed as a product of three Fibonacci numbers. More precisely, our main result shows that the only solution to the Diophantine equation FkFlFm=Jn for 2<k<l<m is (k,l,m,n)=(5,7,8,12). The proof relies on techniques involving linear forms in logarithms.
arXiv:2606.25826v2·Updated Jul 22, 2026·Originally submitted Jun 24, 2026
Complex systems, from gene regulatory networks to neural circuits and ecological food webs, exhibit rich functional behaviour that topology alone does not capture. Yet functional complexity remains difficult to quantify independently of structural organisation. Here we introduce a thermodynamic framework in which functional complexity is characterised through the hierarchical organisation of functional memory, quantifying how the influence of past interactions is distributed and progressively compressed across scales. Across thirty-four empirical networks spanning biological, ecological, social, technological, and biophysical systems and several orders of magnitude in size and density, real interaction strengths organise functional memory at greater hierarchical depth than random weight assignment on the same topology in every domain studied. The framework further reveals that functional memory occupies a remarkably low-dimensional space, collapsing onto four recurrent dynamical organisations. Comparisons with null models that selectively perturb weighted transport geometry, mesoscale wiring, and directionality show that these structural ingredients play distinct roles: weighted transport geometry systematically governs memory depth, whereas mesoscale wiring organises memory across scales and directionality modulates the response of the cascade to structural perturbation. The same comparisons provide an operational criterion for determining whether network weights encode functionally meaningful structure beyond topology. These results establish weighted transport geometry as a primary organiser of functional memory and provide a quantitative framework for studying functional complexity in directed weighted networks.
We prove a flexible criterion for fixed price one, applying in particular to higher-rank lattices over local fields and automorphism groups of affine buildings, including lattices in exotic buildings. It also recovers fixed price one for amenable groups. We then establish a locally compact version, and deduce that every product of two noncompact, compactly generated, unimodular locally compact groups, as well as every lattice in such a product, has fixed price one.
In a recent work [Adv. Math. 401 (2022), Paper No. 108318], a central limit theorem was established for the linear combinations of the iterates of a non-rotational inner function fixing the origin. In this paper, we prove the law of iterated logarithm (LIL) in the same setup, with a very mild condition on the coefficients. We also identify the full set of subsequential limit points at the LIL scale. Using the Aleksandrov--Clark decomposition and measure-preserving properties of the inner functions, one can construct a reverse martingale that is close to the linear combinations of inner functions. We prove the LIL for the partial sums of reverse martingale differences under a Feller-type assumption, which then transfers to the linear combinations of iterates of the inner functions.
We establish explicit polynomial bounds on the stability rate and radius of the lamplighter group. This provides the first example of an infinitely presented group with an explicit upper bound on the stability radius, answering a question of the first author, Levit and Vigdorovich. Our approach is based on new dynamical notions, including the analysis of approximately invariant measures. We establish an effective continuous tower decomposition procedure for approximately invariant measures with presence of periodic points. To achieve polynomial bounds we appeal to recent techniques from descriptive combinatorics and distributed LOCAL algorithms.
Gal Binyamini, Dmitry Novikov, Francesco Maria Saettone
math.NTmath.AG11G0911J9314G05
We extend the Pila-Zannier strategy to Drinfeld modules: we prove analogues of the Manin-Mumford theorem for a product of two Drinfeld modules of equal rank, and of the André-Oort theorem for a product of two Drinfeld modular curves. In characteristic zero, several steps of this strategy rest on o-minimality, which has no counterpart over a function field; we replace the counting step by the rigid analytic Pila-Wilkie theorem of Binyamini-Kato, and this appears to be its first arithmetic application. The functional transcendence input, namely an analogue of the Ax-Lindemann theorem in both settings, is established here by an independent point counting argument.
arXiv:2505.15964v3·Updated Jul 21, 2026·Originally submitted May 21, 2025
Antoine Marnat, Nikolay Moshchevitin, Johannes Schleischitz
math.NT11J13
For a real m×n matrix ξ, we consider its sequence of best Diophantine approximation vectors xi∈Zn,i=1,2,3,..., the sequences of its norms Xi=∥xi∥ and the norms of remainders Li=∥ξxi∥. It is known that, in the cases m=1, bad approximability of ξ is equivalent to the boundedness of ratios XiXi+1, while for n=1 bad approximability of ξ is equivalent to the boundedness of ratios Li+1Li. Moreover, carefully constructed example show that in the cases m=1 and n=1 boundedness of ratios Li+1Li and XiXi+1 respectively (the order of ratios changed), does not imply bad approximability of ξ. In the present paper, we study the impact of the boundedness of ratios on Diophantine properties of ξ, in particular, what restrictions it gives for Diophantine exponents ω(ξ) and ω^(ξ). One of our particular results deals with the case m=n=2. We prove that for 2×2 matrices ξ boundedness of both ratios XiXi+1,Li+1Li implies inequality ω^(ξ)≤34 and that this result is optimal. Our methods combine parametric geometry of numbers as well as more classical tools.
arXiv:2410.02032v2·Updated Jul 21, 2026·Originally submitted Oct 2, 2024
We study the complexity of S-adic sequences corresponding to a family of 216 multidimensional continued fractions maps, called Triangle Partition maps (TRIP maps), with an emphasis on those with low upper bounds on complexity. Our main result is to prove that the complexity of S-adic sequences corresponding to the triangle map (called the (e,e,e)-TRIP map in this paper) has upper bound at most 3n. Our second main result is to prove an upper bound of 2n+1 on complexity for another TRIP map. We discuss a dynamical phenomenon, which we call ``hidden R2 behavior,'' that occurs in this map and its relationship to complexity. Combining this with previously known results and a list of counter-examples, we provide a complete list of the TRIP maps which have upper bounds on complexity of at most 3n, except for one remaining case for which we conjecture such an upper bound to hold.
arXiv:2008.09698v2·Updated Jul 21, 2026·Originally submitted Aug 21, 2020
A generalization of the well--known Fibonacci sequence is the k--Fibonacci sequence with some fixed integer k≥2. The first k terms of this sequence are 0,…,0,1, and each term afterwards is the sum of the preceding k terms. In this paper, we find all k--Fibonacci numbers that are concatenations of two repdigits. This generalizes prior results which dealt with the above problem for the particular cases of Fibonacci and Tribonacci numbers.
In this paper, we use a variety of classical and new research methods for ternary exponential Diophantine equations and extensive use of computer calculations to study the conjecture of R. Scott and R. Styer which asserts that for any fixed relatively prime positive integers a,b and c all greater than 1 there is at most one solution to the equation ax+by=cz in positive integers x,y and z, except for listed specific cases. Precisely, we confirm that for any fixed prime c of the form 2r⋅3+1 with some positive integer r the conjecture holds true, except for only finitely many cases all of which can be effectively determined. Most importantly we prove the conjecture to be true whenever c=7,13, or 97, giving another proof of the result of T. Miyazaki and I. Pink for c=13. We also contribute to the estimation of the number of positive integer solutions (x,y) to the equation ax−by=c for any fixed positive integers a,b and c with both a and b greater than 1. Further, based on a key idea in the proofs of the above results, we present a new application of the Euclidean algorithm for polynomials to the polynomial-exponential Diophantine equation Xm−Xn=qy1−qy2 in positive integers X,y1 and y2, where m and n are given positive integers with m>n, and q is a given prime.
arXiv:2502.00731v2·Updated Jul 21, 2026·Originally submitted Feb 2, 2025
Diophantine approximation explores how well irrational numbers can be approximated by rationals, with foundational results by Dirichlet, Hurwitz, and Liouville culminating in Roth's theorem. Schmidt's subspace theorem extends Roth's results to higher dimensions, with profound implications to Diophantine equations and transcendence theory. This article provides a self-contained and accessible exposition of Roth's theorem and Schlickewei's refinement of the subspace theorem, with an emphasis on proofs. The arguments presented are classical and approachable for readers with a background in algebraic number theory, serving as a streamlined, yet condensed reference for these fundamental results.
arXiv:2603.25814v3·Updated Jul 20, 2026·Originally submitted Mar 26, 2026
We continue our investigations of bilinear sums with modular square roots and the large sieve for square moduli in our recent article "On bilinear sums with modular square roots and applications II", arXiv:2603.00768. In the present article, we focus on the case of prime square moduli for which our previous method in the said article did not yield any improvement. Now we modify this method to make progress for these moduli. The key idea is to restrict certain quadratic Gauss sums to reduced residue classes, which results in significant cancellations in certain cases.
arXiv:2603.00768v3·Updated Jul 20, 2026·Originally submitted Feb 28, 2026
In this article, we continue our recent investigations on bilinear sums and additive energies with modular square roots. Here we improve our recent results for the case when the ranges of variables are large. We use these results to make further partial progress on the large sieve for square moduli.
For p∈(0,1), sample a binary sequence from the infinite product measure of Bernoulli(p) distributions. It is known that for p=1/2, almost every binary sequence is Poisson generic in the sense of Peres and Weiss, a property that reflects a specific statistical pattern in the frequency of finite substrings. However, this behaviour is highly exceptional: it fails for any p=1/2. In these other cases, we show that the frequency of substrings of almost every sequence has either trivial or peculiar behaviour. Nevertheless, the Poisson limiting regime can be recovered if one restricts attention to substrings with a fixed number of successes in the Bernoulli(p) trials.
We study Heisenberg uniqueness for the positive hyperbola branch and shifted lattice crosses in the supercritical regime q=αγ>1. We resolve the infinite-dimensionality clause of the arbitrary-shift problem posed by Giri and Manna: for arbitrary shifts on both arms, the normalized pre-annihilator is infinite-dimensional. More precisely, every v∈BV((1,q)) has a global BV pre-annihilating extension; the extension is unique unless both twisting phases are trivial, in which case its ambiguity is one-dimensional. The proof reduces the annihilation conditions to a graph equation for a twisted Perron--Frobenius operator and combines a phase-uniform Lasota--Yorke estimate with peripheral spectral rigidity. We also give an exact operator-theoretic normal form for the entire L1 pre-annihilator in terms of the maximal convergence domain of the associated Green series. Writing Q for the twisted product and A for the forcing operator, we show that Q has the closed unit disk as its spectrum on L1((0,1)), that Ran(I−Q) is not closed, and that, outside a countable set of algebraic values of q>1, the operator ∑j=0N−1QjA:L1((1,q))→L1((0,1)) has norm 2N for every N≥1.
We present an expository proof of the irrationality of ζ(3) using modular forms of level 6. By constructing a suitable Eichler integral, we obtain a power series with controlled denominators and sufficiently large radius of convergence. Beukers' irrationality criterion then implies that ζ(3) is irrational.
Let G be a connected semisimple real Lie group, Γ an irreducible lattice in G and X=G/Γ. Let F={gt:t≥0} be a non-quasiunipotent one-parameter subsemigroup of G. Then it is known that the set of points in X with bounded F-trajectories has full Hausdorff dimension. In addition, if U is the expanding horospherical subgroup relative to g1, then for any x∈X the set of points u∈U such that the F-trajectory of ux is bounded has full Hausdorff dimension. In this paper we take U to be a horospherical subgroup of G and apply Shi's equidistribution theorem for elements of the expanding cone with respect to U to describe a class of subsets F in G, not presupposing the group structure, for which the above full Hausdorff dimension statements also hold. As an application, we prove that the set of badly approximable matrices in the set-up of Diophantine approximations with quasimultiplicative weight functions has full Hausdorff dimension.
arXiv:2605.14574v2·Updated Jul 19, 2026·Originally submitted May 14, 2026
We use normal-turn estimates to study the global and local geometry of the boundaries of McShane--Rivin norm balls BX for complete finite-area hyperbolic once-punctured tori X. This yields a logarithmic-square bound for the number of integer points on the boundary of each dilated norm ball. Consequently, the number of simple closed geodesics of length exactly L≥2 is at most CX(logL)2. For the modular torus, this gives #λM−1(m)≤C(loglog(3m))2 for every Markoff number m, improving the previous logarithmic bounds for Markoff fibers. Our second result shows that the boundary ∂BX is a convex-geometric detector of exponential Diophantine approximation: a rational direction gives genuine corner with exponentially small exterior angle in the hyperbolic length of the corresponding simple closed geodesic, while at an irrational direction β the graph-flatness order admits an explicit formula in terms of the exponential rate at which rational directions approach β and the ℓ∞-radius of BX in the projective direction β. Thus, irrational directions are not uniformly flat to infinite order, correcting the McShane--Rivin local picture. We also determine all possible irrational flatness orders and the size of the corresponding level sets; in particular, every intermediate finite-flatness level determines the marked torus.
We study entropy density for full shifts over amenable residually finite groups. Entropy density means that every invariant measure can be approximated in the weak∗ topology, together with its entropy, by measures from a distinguished family. In symbolic dynamics it is known, by a result of Weiss, that uniquely ergodic measures are entropy dense among ergodic measures for the shift action of Z. We extend this result to full shifts over amenable residually finite groups: invariant measures supported on uniquely ergodic subsystems are entropy dense in the collection of ergodic invariant measures. This applies in particular to full shifts over finitely generated abelian groups. The proof uses Cortez--Petite Følner tilings and a block-replacement construction adapted to finite-index subgroups.
arXiv:2508.17193v2·Updated Jul 18, 2026·Originally submitted Aug 24, 2025
Let Sg be the closed surface of genus g, L be the infinite Jacob's ladder surface, and Map(S) denote the mapping class group of a surface S. Let qg:L→Sg be the regular infinite-sheeted cover with deck transformation group Z. In this paper, we show the existence of ``pseudo-Anosov-like'' maps on L that arise as the lifts of Penner-type pseudo-Anosov maps on Sg under the cover qg. Furthermore, we establish that these lifts are topologically transitive, mixing, and support null recurrent dynamics. Moreover, we present concrete examples of infinite families of such maps on L.
arXiv:2402.02660v4·Updated Jul 17, 2026·Originally submitted Feb 5, 2024
Ramanujan studied a general class of Stirling constants that are the resummation of some natural divergent series. These constants include the classical Euler-Mascheroni, Stirling and Glaisher-Kinkelin constants. We find natural integral representations for all these constants that appear as exponential periods in the field Q(t,e−t) which reveals their natural transalgebraic nature. We conjecture that all these constants are transcendental numbers. Euler-Mascheroni's and Stirling's integral formula are classical, but the integral formula for Glaisher-Kinkelin appears to be new, as well as the integral formulas for the higher Stirling-Ramanujan constants. The method presented generalizes naturally to prove that many other constants are exponential periods over the field Q(t,e−t).
This work studies the mean-field limit of large networks of interacting stochastic leaky integrate-and-fire (LIF) neurons subject to short-term synaptic depression (STD). The macroscopic dynamics of this system is governed by a two-dimensional, non-linear McKean-Vlasov equation that couples the evolution of the neurons' membrane potentials with a synaptic depression variable. We investigate the long-time behavior of this limit system. To this end, we introduce an auxiliary linearized Markov process by freezing the interaction non-linearity to a constant. By exploiting the regeneration of the membrane potential at spike times, we are able to explicitly compute the conditional expectation of the synaptic depression variable, conditionally on the potential value, under the invariant measure of this two-dimensional linear process. This is a crucial ingredient to study time-dependent local perturbations thereof. As a consequence we are able to identify an analytic criterion guaranteeing the local stability of any invariant probability measure of the fully non-linear system. This stability criterion is formulated in terms of the zeros of the Laplace transform of a specific linear response function. Finally, we provide numerical examples demonstrating that the two-dimensional framework induces a richer spectrum of long-time dynamics than purely one-dimensional models. For example, synaptic depression can lead to low-frequency oscillations around a unique, unstable invariant measure where the oscillations are much slower than the neurons' firing rates.
Motivated by Sarnak's conjecture in topological dynamics for the Möbius function μ, we study, for a power-bounded T on a Banach space E, the weak convergence (∗)N1n=1∑Nμ(n)Tnv→0 weakly ∀v∈E. For that, we introduce a notion of dynamical entropy for operators, which we denote htop∗(T), and show that if Sarnak's conjecture is true, then htop∗(T)=0 implies the desired convergence (*). We conclude an equivalent operator formulation of Sarnak's conjecture. For several classes of operators we prove that (*) holds, and that htop∗(T)=0.
arXiv:2601.17988v2·Updated Jul 16, 2026·Originally submitted Jan 25, 2026
We prove that for an arbitrary indexing group, every ergodic infinitely divisible stationary process that is separable in probability is weakly mixing. This shows that, as in the well-known case of Gaussian stationary processes, ergodicity implies weak mixing is intrinsic to infinite divisibility, removing all structural assumptions on the group from prior results. The main ingredient is a general construction of stochastically continuous extensions for separable in probability stationary processes, reducing the problem to stochastically continuous processes indexed by Polish groups and then to countable groups, where we combine the Maruyama representation with an ergodicity criterion for Poisson suspensions.
arXiv:2405.17122v2·Updated Jul 16, 2026·Originally submitted May 27, 2024
Tattwamasi Amrutam, Martin Klötzer, Hanna Oppelmayer
math.DSmath.FA37A5022F1037B0560J50
Let G be a locally compact second countable group equipped with an admissible non-degenerate Borel probability measure μ. We generalize the notion of μ-stationary systems to μ-stationary G-factor maps π:(X,ν)→(Y,η). For these stationary relations between dynamical systems, we provide a structure theorem, which generalizes the structure theorem of Furstenberg-Glasner. Furthermore, we show the existence and uniqueness of a relative version of the Poisson boundary in this setup.
In this paper we introduce and explore the notion of rigidity group, associated with a collection of finitely many sequences, and show that this concept has many, somewhat surprising characterizations of algebraic, spectral, and unitary nature. Furthermore, we demonstrate that these characterizations can be employed to obtain various results in the theory of generic Lebesgue-preserving automorphisms of [0,1], IP-ergodic theory, multiple recurrence, additive combinatorics, and spectral theory. As a consequence of one of our results we show that given (b1,...bℓ)∈Nℓ, there is no orthogonal vector (a1,…,aℓ)∈Zℓ with some ∣aj∣=1 if and only if there is an increasing sequence of natural numbers (nk)k∈N with the property that for each F⊆{1,...,ℓ} there is a μ-preserving transformation TF:[0,1]→[0,1] (μ denotes the Lebesgue measure) such that for any measurable A,B⊆[0,1], k→∞limμ(A∩TF−bjnkB)={μ(A∩B), if j∈F,μ(A)μ(B), if j∈F. We remark that this result has a natural extension to a wide class of families of sequences.
We prove an all-directions Marstrand-Mattila projection theorem for self-affine measures and sets in Rd. Under exponential separation, together with proximality and strong irreducibility assumptions on the linear parts, the projection of a self-affine measure onto every line has the expected Hausdorff dimension. If the proximality assumption is strengthened to strong pinching, then the same conclusion holds for the self-affine set X itself, without any separation assumption. In the plane, strong irreducibility of the linear parts alone suffices, and this is sharp. As a corollary, if X additionally has upper Minkowski dimension at most one, then its Minkowski dimension exists and equals its Hausdorff dimension, giving a partial affirmative answer to the folklore question of whether the Minkowski dimension exists for every self-affine set.
We prove the analogue of Belinskaya's theorem for measure-preserving flows: two free ergodic measure-preserving flows whose L1 full groups are isomorphic as abstract groups are conjugate up to a scalar time change. This answers a question posed by François Le Maître and the author. We show that whenever two free ergodic flows generate the same orbit equivalence relation and one is contained in the other's L1 full group, their positive half-orbits are commensurate after possibly reversing time. Katznelson's criterion then yields conjugacy after a scalar time change. The key new ingredient is a commensuration criterion asserting that a measurable subset of the real line whose symmetric differences with its translates have finite average measure over the unit interval is commensurate with exactly one of the empty set, the whole line, and the two half-lines. This criterion and its application were discovered autonomously by a two-agent AI system. The author independently verified the proofs and prepared the final text.
arXiv:2601.10051v4·Updated Jul 15, 2026·Originally submitted Jan 15, 2026
We present the Conservative Matrix Field (CMF) as a tool for the analysis and computation of D-finite functions. We use conservative matrix fields to establish asymptotic properties of families of linear forms in periods, including (but not limited to) multivariate Mellin integrals, via a discrete Levinson-type framework due to Benzaid and Lutz. Finally, we present an experimental analysis of the families of linear forms generated by these objects and formalize the resulting observations as conjectures on their continuous asymptotic and arithmetic properties.
Motivated by the Lester-Wigman vanishing area correlation conjecture for lattice points near the boundary of circles of growing radius, we investigate the dynamics of circle flows on tori (which are related to the motion of a charged particle in a magnetic field on a torus.) We show that an analogue of the vanishing correlation conjecture holds in this setting, i.e., we have "mixing for the area observable" despite the flow being essentially integrable. We also determine the probability density function of the areas, in global as well as local regimes.
We study the concatenated Fibonacci constant F:=0.F1F2F3⋯=0.11235813⋯, obtained by concatenating the Fibonacci numbers in the fractional part, and ask whether it is normal. We show that several classical sufficient conditions for normality by concatenation do not apply to the Fibonacci sequence because of its exponential growth, while a criterion of Pollack and Vandehey implies that the normality of F in base 10 would follow if almost all Fibonacci numbers were (ε,k)-normal in base 10. The Benford bias of leading digits and the Pisano periodicity of trailing digits are shown to contribute asymptotically negligible fractions of the total digits, isolating the distribution of the deep digits of large Fibonacci numbers as the remaining obstruction. Large-scale numerical experiments on the first 500,000 Fibonacci numbers in bases 10 and 2 indicate that global single-digit counts and k-block statistics for k=2,3,4 are compatible with i.i.d.-like fluctuations at the scales tested, and that a positional decomposition concentrates the visible structured deviation at the boundaries between consecutive Fibonacci numbers, while pooled interior blocks remain close to uniform. Our computations suggest that any obstruction to normality lies in the asymptotic behavior of the deep digits of Fn.
We construct an example of a random dynamical system on the circle, formed by maps that are only locally invertible, which possesses an atomic stationary measure ν. Moreover, this measure is physical: for Lebesgue-almost every initial point x0, the Cesàro averages of its random trajectory almost surely converge to ν. This shows that the Hölder regularity of stationary measures, known for (non-measure-preserving) random dynamical systems formed by diffeomorphisms, cannot be generalized to this class of systems. We also provide some related examples, including ones where a stationary measure charges a proper submanifold, despite the absence of a closed common invariant submanifold.
arXiv:2110.02569v4·Updated Jul 13, 2026·Originally submitted Oct 6, 2021
Let Fq be the finite field with q elements and consider the rational function field K:=Fq(θ). For a Drinfeld module φ defined over K, we study the transcendence of special values of the Goss L-function attached to the abelian t-motive Mφ of φ. Moreover, when φ is a Drinfeld module of rank r≥2 defined over K which has everywhere good reduction, we prove that the value of the Goss L-function attached to the (r−1)-st exterior power of Mφ at any positive integer is transcendental over K.
arXiv:2503.16330v2·Updated Jul 13, 2026·Originally submitted Mar 20, 2025
Laura Capuano, Sara Checcoli, Marzio Mula, Lea Terracini
math.NTmath.CO11D8811J7011J8168R15
Continued fraction expansions provide a well-established bridge between algebraic properties of numbers and combinatorics on words. In this article, we investigate the algebraicity of p-adic numbers whose continued fractions arise from certain classes of words which generalize the classical automatic, periodic and palindromic words. Our main result shows that, under mild conditions on the p-adic continued fraction expansion, such numbers are either algebraic of degree at most 2 or transcendental. This result provides an analogue of results of Bugeaud and Adamczewski-Bugeaud in the real setting and extends previous works that were limited to specific choices of p-adic floor functions and less general classes of words.
We prove that every free ergodic measure-preserving action of a countably infinite amenable group with positive Rokhlin entropy has infinite complex L1-orbit multiplicity, both on L1 and on its mean-zero subspace L01. This gives a negative answer to a question of J.-P. Thouvenot recorded by Iwanik and establishes the corresponding endpoint statement at p=1 of Iwanik's theorem that positive entropy implies infinite Lp-multiplicity for every p>1. The proof combines Malykhin's rigidity theorem for independent random variables, Seward's Bernoulli factor theorem, and a Følner set argument.
The goal of this note is to systematically develop the fundamental arithmetic and combinatorial properties of the sector renormalization operation on rigid rotations. We employ the specific framework of modified continued fractions appropriate for sector renormalization and analyze their properties. By allowing infinite first return times, this framework yields a dynamical compactification characterized by a universal property. We also discuss the corresponding natural extension and introduce the notion of a time (semi-)group. For example, we demonstrate how a bi-infinite tower of sector renormalizations of irrational rotations can be packaged within a single dynamical plane as a cascade of translations. This note will serve as a foundational combinatorial tool for studying the geometric properties of sector renormalizations of holomorphic maps with irrationally indifferent fixed points, particularly neutral quadratic polynomials.
René Pfitscher, Anurag Rao, Shucheng Yu, Han Zhang
math.DSmath.NT37A1711J1311H06
We establish two distinct zero-one laws for the uniform Diophantine approximation of complex numbers by quotients of Gaussian integers and by quotients of Eisenstein integers. Using tools from homogeneous dynamics, we study this problem by reducing to a shrinking target problem on certain homogeneous spaces of SL2(C). The main novel ingredients include measure estimates on a certain family of neighborhoods of the corresponding critical loci, as well as new disjointness statements to control the short-range mixing contribution. Due to the different nature of the critical loci in the Gaussian and Eisenstein cases, these measure estimates are obtained by rather different arguments.
Let G be a locally compact Abelian (LCA) group with dual group Γ, and let μ be a probability measure on (the Borel sets of) G. Given a unitary representation {U(t):t∈G} in a complex Hilbert space H, we study the spectrum of the μ-average V:=∫GU(t)dμ(t) (defined in the strong topology of H). We prove that σ(V)⊂μ^(Γ) and give a sufficient condition for equality. Using the spectral measure E(⋅) given by the general Stone theorem, we prove a (weak) spectral mapping theorem for the operators U(ν):=∫GU(t)dν(t), where ν is any bounded complex measure on G. For a unitary representation of Z, defined by the powers of a unitary operator U, we prove that σ(V)=μ(σ(U)). For a unitary representation of R, given as U(t)=eitB (t∈R), we show that σ(V)=μ(σ(B)).
arXiv:2211.07474v2·Updated Jul 13, 2026·Originally submitted Nov 14, 2022
In this paper, we obtain global function field versions of the results of Schinzel-Postnikova for multiplicative groups, and of Hahn-Cheon for elliptic curves, which is an analog of the former result.
We prove effective equidistribution of expanding horocycles in SL2(Z)\SL2(R) with respect to various classes of Borel probability measures on R having certain Fourier asymptotics. Our proof involves new techniques combining tools from automorphic forms and harmonic analysis. In particular, for any Borel probability measure μ, satisfying ∑Z∋∣m∣≤X∣μ(m)∣=O(X1/2−θ) with θ>7/64, our result holds. This class of measures contains convolutions of s-Ahlfors regular measures for s>39/64, as well as a subclass of self-similar measures. Moreover, our result is sharp upon the Ramanujan--Petersson Conjecture (upon which the above θ can be chosen arbitrarily small): there are measures μ with μ(ξ)=O(∣ξ∣−1/2+ε) for which equidistribution fails.
arXiv:2006.10821v2·Updated Jul 12, 2026·Originally submitted Jun 18, 2020
We show that a translation bounded measure has pure point diffraction if and only if it is mean almost periodic. We then go on and show that a translation bounded measure has pure point diffraction and satisfies the so called consistent phase property if and only if it is Besicovitch almost periodic. Finally, we show that a translation bounded measure has pure point diffraction and satisfies the consistent phase property independent of the underlying van Hove sequence if and only if it is Weyl almost periodic. These results solve fundamental issues in the theory of pure point diffraction and answer questions of Lagarias.
We give 39 rapidly convergent continued fractions for Chowla--Selberg gamma quotients, and deduce good irrationality measures for 20 of them, including for CS(−3)=(Γ(1/3)/Γ(2/3))3, for a1/4CS(−4)=a1/4(Γ(1/4)/Γ(3/4))2 with a=12 and a=1/5, and for CS(−7)=Γ(1/7)Γ(2/7)Γ(4/7)/(Γ(3/7)Γ(5/7)Γ(6/7)). These appear to be the first proved and reasonable irrationality measures for gamma quotients.
This paper computes explicit constants in Bombieri and André's height bound for points at which there are unexpected "global" polymomial relations between values of G-functions. It applies these bounds for G-functions to obtain explicit height bounds for unlikely intersections with lines in tori, making explicit a weak version of the bounded height theorem of Bombieri, Masser and Zannier and, in some cases, improving an explicit bound of Habegger.
arXiv:2510.19711v2·Updated Jul 10, 2026·Originally submitted Oct 22, 2025
Sejal Babel, Melih Emin Can, Dominik Kwietniak, Piotr Oprocha
math.DS37A0537A3037B10
We describe the spectrum of an ergodic invariant measure by examining the behaviour of its generic points. We define regular Wiener--Wintner generic points for a measure to generalise the characterisation of generic points for discrete spectrum measure from Lenz et al. [Ergodic Theory and Dynamical Systems vol. 44 (2024), no. 2, 524--568]. We also study limits of sequences of generic points with respect to the Besicovitch pseudometric. This translates to results about limits of measures with respect to the metric rho-bar ρˉ generalising Ornstein's d-bar metric. We study how the spectrum behaves when passing to the limit and we prove that points generic for discrete spectrum, totally ergodic, or (weakly) mixing measures, property K, zero entropy measures form a closed set with respect to the Besicovitch pseudometric. Hence, the same holds for corresponding measures with respect to the rho-bar metric. Our methods have already been used to prove existence of ergodic measures with desired properties, in particular with discrete spectrum. They also lead to a new proof of rational discrete spectrum of the Mirsky measure associated with a given set of B-free numbers.
arXiv:2402.00739v3·Updated Jul 10, 2026·Originally submitted Feb 1, 2024
Laura Capuano, Marzio Mula, Lea Terracini, Francesco Veneziano
math.NT11J7011D8811D09
Inspired by several alternative definitions of continued fraction expansions for elements in Qp, we study p-adically convergent periodic continued fractions with partial quotients in Z[1/p]. To this end, following a previous work by Brock, Elkies, and Jordan, we consider certain algebraic varieties whose points represent formal periodic continued fractions with period and preperiod of fixed lengths, satisfying a given quadratic equation. We then focus on the p-adically convergent loci of these varieties, characterizing the zero and one-dimensional cases.
We present a novel proof of the Duffin-Schaeffer conjecture in metric Diophantine approximation. Our proof is heavily motivated by the ideas of Koukoulopoulos-Maynard's breakthrough first argument, but simplifies and strengthens several technical aspects. In particular, we avoid any direct handling of GCD graphs and their `quality'. We also consider the metric quantitative theory of Diophantine approximations, improving the (logΨ(N))−C error-term of Aistleitner-Borda and the first named author to exp(−(logΨ(N))21−ε).
The real projective line RP1 is the boundary of HP={z∈C:ℑ(z)>0}, a model of the hyperbolic plane whose space of geodesics identifies with G(HP)=RP1×RP1∖diagonal. The modular group Γ=PSL2(Z) acts on HP with quotient the modular orbifold M=Γ\HP. Consider a finite-index subgroup of the modular group Γ′⊂Γ=PSL2(Z) corresponding to a finite cover M→M′. A geodesic (ξ−,ξ+)∈G(HP) projects modΓ′ to a geodesic ξ′⊂M′. We show that if ξ′ is simple, then ξ+ is either rational or quadratic or transcendental. In the transcendental case, we obtain bounds on the Mahler measures and show that those can be improved for geodesics fixed by pseud-Anosov maps. Finally, we also explain in detail why all this was known for the modular torus cover associated to the derived subgroup Γ′=[Γ,Γ].
Let F=Fq((t−1)),G=SL2(F),Γ=SL2(Fq[t]),X=Γ\G, and let K=SL2(O), where O=Fq[[t−1]]. We study right K-spherical averages along the upper unipotent subgroup, the horospherical subgroup associated with the standard cusp, on the Nagao lattice quotient. The basic observation is that the K-spherical projection converts two natural dynamical families - expanding translates of compact unipotent orbits and cusp-adapted truncations of dense unipotent orbits - into the same rooted descendant problem on the Bruhat--Tits tree. In the even bipartite sector the limiting height law is the explicit probability measure ρev(0)=qq−1,ρev(2m)=(q2−1)q−2m−1(m≥1). We prove an exact discrepancy formula: in the backward state the error is a pure top-shell term minus a missing tail, while in the forward state the error is a first-turn weighted sum of backward errors. These formulas give quantitative K-spherical equidistribution for expanding translates of compact U-orbits and for dense-orbit truncations. For compactly supported K-spherical observables in the expanding translates of compact orbits, the discrepancy is eventually exactly zero. In the dense case the rate is controlled by the continued-fraction expansion of the boundary point attached to the orbit.
arXiv:2510.22484v2·Updated Jul 9, 2026·Originally submitted Oct 26, 2025
In the context of (not necessarily minimal) actions, we consider the mean diameter and use it to characterize regular factor maps. Building on this characterization, we prove that an action is diam-mean equicontinuous if and only if it is a regular extension of its maximal equicontinuous factor. Furthermore, we establish the existence of a maximal diam-mean equicontinuous factor and discuss stability properties of regular factor maps. For this, we work in the context of actions of locally compact and σ-compact amenable groups.
When markets move more and more in lockstep, are they drifting towards the point where a price bubble becomes possible, and can that drift be measured before the crossing? This paper joins two long-separate ideas, that a rational bubble is a price outgrowing its dividends and that a crisis threshold can be read off the strength of a market's single dominant factor, onto one object recovered from the data: a summary of how asset returns move together, paired with a discount rate. We call this crossing point the fragility edge and show it plays three roles at once. A stated discipline says what the data support: the edge firmly, with a margin of error; whether a bubble exists, only roughly; which asset carries it, not at all. Across eighteen global equity indices from 2004 to 2024, that dominant factor strengthens in every documented crisis, the market collapsing from about six to about four independent factors; once the discount is set so that calm markets sit at the edge, this strength crosses it in crisis. These readings coincide with crises, not forecasts.
arXiv:2602.14142v3·Updated Jul 9, 2026·Originally submitted Feb 15, 2026
We study the Reverse algorithm, a multidimensional continued fraction algorithm, which is not unimodular. We show that the Reverse algorithm is ergodic and, by proving that its second Lyapunov exponent is negative, that it is a.e. exponentially convergent. In addition to that, we attach substitutions to this algorithm and study the S-adic languages generated by sequences of these substitutions. The negativity of the second Lyapunov exponent implies that almost all of these languages are balanced. By a thorough study of the combinatorics of the substitutions, we are even able to obtain a concrete generic family of balanced languages that is characterized in terms of a simple condition on the underlying sequence of substitutions.
We show that a fully inhomogeneous uniform Littlewood type problem has a negative answer and the counterexamples form a dense Gδ set. This extends the author's recent analogous result for the homogeneous case. The main difficulty in the general setting is the semi-homogeneous case where one linear form is homogeneous and the other inhomogeneous with irrational shift. We further address the higher dimensional analogue.
We consider rotational beta expansions in dimensions 1, 2 and 4 and view them as expansions on real numbers, complex numbers, and quaternions, respectively. We give sufficient conditions on the parameters α,β∈(0,1) so that particular cylinder sets arising from the expansions are winning or losing Schmidt (α,β)-game.
arXiv:2410.18554v2·Updated Jul 8, 2026·Originally submitted Oct 24, 2024
Jeroen S. W. Lamb, Guillermo Olicón-Méndez, Martin Rasmussen
math.PRmath.DS37H2037A5037C3060J0560G10
We study the asymptotic behaviour of stationary densities of one-dimensional random diffeomorphisms, at the boundaries of their support, which correspond to deterministic fixed points of extremal diffeomorphisms. In particular, we show how this stationary density at a boundary depends on the underlying noise distribution, as well as the linearisation of the extremal diffeomorphism at the boundary point (in case the corresponding fixed point is hyperbolic), or the leading nonlinear term of the extremal diffeomorphism (in case the corresponding fixed point is not hyperbolic).
arXiv:2508.18044v6·Updated Jul 8, 2026·Originally submitted Aug 25, 2025
Recently, the authors showed that for every irrational number α, there exist infinitely many positive integers n represented by any given positive definite binary quadratic form Q, satisfying ∣∣αn∣∣<n−(1/2−ε) for any fixed ε>0. We also provided a quantitative version with a lower bound when the exponent 1/2−ε is replaced by a smaller exponent γ<3/7−ε. In this article, we establish a quantitative version for the exponent 1/2−ε, where we confine ourselves to the particular case of sums of two squares.
Let p be a prime number. We introduce a sparseness condition on the supports of p-adic Hahn series, and prove that this condition implies transcendence over Q˘p, the completed maximal unramified extension of Qp. As an application, we prove the order-type conjecture of Qp-algebraic p-adic Hahn series with bounded support under the condition that the support has only finitely many accumulation points. All results in this paper have been fully formalized in the Lean theorem prover (v 4.31.0), building over Mathlib.
The problem of classifying geometric structures on manifolds is very much related to the discussion of the automorphism groups actions on character varieties, which are spaces of equivalence classes of representations. In this chapter we survey some results on this topic, mostly focusing on representations of surface groups (both in the orientable and non-orientable cases) and free groups. An important principle in the study of the dynamics on character varieties X=X(π1(S),G) for surface groups π1(S) is the following dichotomy: when the target group G is compact, X has nontrivial homotopy type, and the action of the mapping class group is chaotic; whereas when the target group G is non-compact, X contains contractible sets on which the mapping class group acts properly. We will expand on this dichotomy in various cases. After introducing the necessary background, we will discuss representations into PSL2(R) and PGL2(R), discussing the number of connected components, the geometric properties (Bowditch question), the dynamics (Goldman conjecture) and some components with an `exotic' behaviour (Deroin-Tholozan representations). We will also underline how the theory for representations of fundamental groups of orientable closed hyperbolizable surfaces needs to be adapted when one considers surfaces with punctures or non-orientable surfaces. We will then discuss representations into compact groups, where we will discuss mostly ergodicity results in various settings, and some non-ergodicity results at the end. Thirdly, we will consider representations in PSL2(C). We will discuss convex-cocompact representations, primitive-stable and Bowditch representations and their relationship. Finally, we will describe how some of the results mentioned can be generalized for representations into higher-rank Lie groups.
arXiv:2501.06380v4·Updated Jul 7, 2026·Originally submitted Jan 10, 2025
In this note, we investigate coboundaries of interval exchange transformations of three intervals, or 3-IETs. More precisely, we study differentiable functions whose derivative is absolutely continuous, and whose integral is zero and whose derivative also has integral zero. We show that these functions are coboundaries for a typical 3-IET if and only if their values at the endpoints of the domain are zero. We also show the existence of exceptional counterexamples for both possible endpoint behaviors. Our results are obtained by studying the properties of associated skew products.
arXiv:2606.27034v2·Updated Jul 7, 2026·Originally submitted Jun 25, 2026
Let C be the middle-third Cantor set and let μ be the natural Cantor probability measure. Let γ=log3log2. The two main results of this paper are μ{x∈C:∥2nx∥<n−τ for infinitely many n}=0 for τ>2−γ. and μ{x∈C:∥2nx∥<n−τ for infinitely many n}=1 for τ<21−γ. These results give new progress toward Velani's conjecture on zero-one law for dyadic approximation in the middle-third Cantor set.
We study quantitative time averages of spherical observables sampled along equidistributed spirals generated by Kronecker flows. For a Diophantine frequency vector and Holder observables on the sphere, we prove a window-uniform polynomial convergence rate toward the spherical mean. The main difficulty is that the natural measure-preserving parametrization of the sphere is not continuous on the torus and has polar degeneracies; we handle this through a localization and de la Vallee Poussin approximation scheme. We then apply the estimate to shrinking spirals converging to a center, obtaining explicit rates for spherical means under mild radial regularity. We also treat integrable angular data and homogeneous singularities at the center, proving a dichotomy between power-law and exponentially shrinking radial profiles. Finally, we show that no uniform rate can hold over all rationally independent frequencies, even for smooth mean-zero data on the sphere.
arXiv:2606.29352v2·Updated Jul 5, 2026·Originally submitted Jun 28, 2026
In this paper, we study the long-time behaviour of the two-dimensional stochastic damped Euler equation on the torus driven by bounded random forcing. Unlike stochastic Navier-Stokes or fractionally dissipative Euler equations, the model possesses no viscous regularisation, so the classical parabolic smoothing is unavailable. We prove that when the damping coefficient is sufficiently large, the associated Markov semigroup admits a unique invariant measure and converges exponentially fast to equilibrium. The key ingredient is the establishment of a global-in-time W1,∞ estimate for the vorticity. This estimate yields a compact absorbing mechanism in C(T2), which enables us to establish the uniqueness of the invariant measure and exponential mixing. To the best of our knowledge, this is the first exponential mixing result for a genuinely inviscid stochastic Euler-type equation. Our approach demonstrates that sufficiently strong linear damping can effectively replace the compactness mechanism usually provided by viscosity and is expected to be applicable to other inviscid or weakly dissipative stochastic partial differential equations driven by bounded random forcing.
arXiv:2601.21209v2·Updated Jul 5, 2026·Originally submitted Jan 29, 2026
J.~Rosen introduced the ring PA0 of so-called finite algebraic numbers, which may be seen as an analogue of certain periods in the ring A=∏pZ/pZ/⨁pZ/pZ, p running through all prime numbers. In this article, we introduce its positive characteristic analogue PAK0 over the rational function field K=Fq(θ), q being a prime power, and study foundational properties, and provide further scopes.
arXiv:2205.09205v2·Updated Jul 5, 2026·Originally submitted May 18, 2022
In this paper we study the action of a countable group Γ on the space of orders on the group. In particular, we are concerned with the invariant probability measures on this space, known as invariant random orders. We show that for any countable group the space of random invariant orders is rich enough to contain an isomorphic copy of any free ergodic action, and characterize the non-free actions realizable. We prove a Glasner-Weiss dichotomy regarding the simplex of invariant random orders. We also show that the invariant partial order on SL3(Z) corresponding to the semigroup of positive matrices cannot be extended to an invariant random total order. We thus provide the first example for a partial order (deterministic or random) that cannot be randomly extended.
Romik has presented a construction of a 1-dimensional dynamical system on the unit interval by developing an algorithm that returns the unique sequence of matrices associated with a positive primitive Pythagorean triple (in the sense of Barning), and projecting the map involved in this algorithm onto an appropriate 1-dimensional space via stereographic projection. Romik additionally computes the infinite, absolutely continuous invariant measure, and shows that the system is conservative and ergodic. Later, Cha et al. provided a method of calculating "Berggren trees", which are generalisations of the tree of positive primitive Pythagorean triples one may construct via Barning's theorem, except for different homogeneous quadratic equations in 3 variables. We present here a method of computing 1-dimensional dynamical systems induced from these Berggren trees following Romik's outline, and determine their absolutely continuous invariant measures by adapting the method of Keane.
We establish L2-exponential strong ergodicity (strong mixing) with an explicit rate of convergence for a class of degenerate diffusions with multiplicative noise and with singular drift in both the noisy and noise-free components. This class includes diffusions with an additional inert drift given by the gradient of a singular potential, as well as singular generalized stochastic Hamiltonian systems. Cases in which the diffusion is confined to a proper, bounded or unbounded subset of Rd1+d2 are included. Concrete examples of admissible potentials are provided. To obtain these results, we use an analytical approach and study the long-time behavior of the strongly continuous contraction semigroup generated by the formal Kolmogorov backward operator. Using the theory of generalized Dirichlet forms, these objects are then identified with the transition semigroup and generator of the unique weak solution to the original stochastic differential equation. The existence and uniqueness of this solution are established under near-minimal conditions.
We establish a relativised variational principle for the Feng--Huang weighted topological entropy associated with a factor map between dynamical systems. Combined with a recent theorem of Yin, this yields an almost-everywhere equivalence between the Feng--Huang entropy and its combinatorial version on fibers. As an application, we compute the Hausdorff dimension of the intersection of random translates of two Bedford--McMullen carpets. The resulting formula extends the Kenyon--Peres formula from the self-similar to the self-affine setting, and also points to a new problem concerning random matrix products.
We determine the minimal absolute value of a non-vanishing sum of n fifth roots of unity chosen with repetition, and characterize the corresponding sums. As a function of n, the minimal absolute value is monotone non-increasing over congruence classes of n modulo 5 and its only jumps occur when n=5Fm, n=Lm, or n=2Lm, where Fm and Lm denote the m-th Fibonacci and Lucas numbers respectively. To prove our results we reduce the problem to a series of inequalities involving rational approximations of the golden ratio φ=(1+5)/2, the solutions of which can be characterized using the theory of continued fractions.
We show the existence of a topologically free minimal action of F∞ on the Cantor space that does not have dynamical comparison. Moreover, we show that this phenomenon can happen both in the presence and in the absence of invariant measures. We also show that strict comparison of the reduced crossed product C*-algebra does not imply dynamical comparison for minimal actions. Our technique involves constructing a monoid which is not almost unperforated, embedding it into a countable refinement monoid and then realising it as the type semigroup associated to a dynamical system.
We consider large Hermitian matrices whose entries are defined by evaluating the exponential function along orbits of the skew-shift 2j(j−1)ω+jy+xmod1 for irrational ω. We establish a rigorous connection between the effective Ratner equidistribution theorem for unipotent orbits in SL(3,R)/SL(3,Z) and the global semicircle law for such deterministic matrices. For frequency sequences satisfying a Diophantine condition, we prove that the empirical spectral distribution of these matrices converges to the Wigner semicircle law with optimal polynomial rate O(N−1); for rectangular matrices the corresponding Marchenko--Pastur law is obtained. The proof uses a multi-parameter effective mixing property derived from the effective Ratner equidistribution theorem, combined with a graph-theoretic expansion of the moments. Our results evidence the quasirandom nature of the skew-shift dynamics observed in other contexts by Bourgain, Goldstein and Schlag, and Rudnick, Sarnak and Zaharescu, and provide a dynamical systems proof of the semicircle law with an improved convergence rate.
We show power Fourier decay for equilibrium states of nonlinear, area preserving, smooth Axiom-A diffeomorphisms on surfaces. This implies positivity of the lower Fourier dimension for self-conformal measures under C1+ iterated function systems that are factors of hyperbolic diffeomorphisms, which is the first result of this kind in this low-regularity setting. To do so, we use the sum-product phenomenon to reduce Fourier decay to the study of a temporal distance function for a well chosen suspension flow, behaving like a 3-dimensional Axiom A flow, whose mixing properties reflects the nonlinearity of our base dynamics. We then generalize in an Axiom A setting the methods of Tsujii-Zhang, dealing with exponential mixing of three-dimensional Anosov flows arXiv:2006.04293. The nonlinearity condition is generic and can be checked in concrete contexts. To illustrate the applications, we prove two corollaries. We first establish a spectral gap, proving exponential mixing for generic circle extensions over hyperbolic maps on surfaces. As a second application, we prove power Fourier decay for the density of states measure of the Fibonacci Hamiltonian. This implies phase-averaged escape-of-mass estimates, which is the first result of this type in a quasicrystal.
A repdigit is a positive integer that has only one distinct digit in its decimal expansion, i.e., a number has the form d(10m−1)/9 for some m≥1 and 1≤d≤9. Let (Tn)n≥0 be the Tribonacci sequence. This paper deals with the presence of repdigits in the products of consecutive shifted Tribonacci numbers.
We study directional expansion for probability-measure-preserving actions of countable groups through a representation-theoretic group property, the cyclic escape property. An infinite countable group has the cyclic escape property if every totally ergodic unitary representation has arbitrarily small fixed-vector projections along infinite cyclic subgroups. This property implies directional expansivity for all totally ergodic actions. We prove that all infinite finitely generated nilpotent groups have the cyclic escape property, and conjecture the same for all infinite finitely generated polycyclic groups. We also prove the cyclic escape property for higher-rank simple lattices whose finite-dimensional unitary representations all have finite image; in particular, for SLn(Z), PSLn(Z), and PGLn(Z), n≥3. By contrast, free groups of rank at least two do not have the cyclic escape property. The proofs exhibit two independent mechanisms: central spectral structure in nilpotent groups and stationary character rigidity in higher-rank lattices.
arXiv:2605.21098v2·Updated Jul 1, 2026·Originally submitted May 20, 2026
In 2008, Dan Romik studied in this journal Primitive Pythagorean Triples, or PPTs. In order to do so, he introduced a modified slow (subtractive) Euclidean algorithm, and showed that the underlying dynamical system of this Euclidean algorithm (the ``Romik system''), is ergodic and has a σ-finite, infinite measure, of which is explicitly given. In this paper, the Romik system is further studied. Various basic properties are determined, such as the expansion of rational numbers and quadratic irrationals. Also (a version of) the planar natural extension of the Romik system is obtained, and the σ-finite, invariant measure is explicitly given, and it is shown that it is ergodic. Furthermore, for Lebesgue almost every x asymptotically half of the regular continued fraction (RCF) convergents of x are among the Romik convergents. We also show that related to the Romik map a ``strange'' continued fraction can be given. ``Strange,'' as the set of possible partial quotients (i.e., digits) for any x∈[0,1] in this expansion is {0,±2}. Various properties of this ``Romik expansion'' are given.
arXiv:2509.00898v2·Updated Jun 30, 2026·Originally submitted Aug 31, 2025
In this note we study the distribution of the intersections between certain translates of closed orbits of the positive diagonal subgroup in SL(3,Z)\SL(3,R) with a maximal parabolic subgroup. These intersections are closely connected to roots of congruences for certain monic, irreducible cubic polynomials F(X)∈Z[X]. The the main result is that the intersections, considered as a sequences in the diagonal subgroup and the parabolic subgroup, are jointly equidistributed. This implies that certain affine lattices determined by pairs of roots of the cubic congruences are jointly equidistributed with corresponding ideals in the associated ring of integers. We note that the techniques here roughly parallel those which has been developed to study the multidimensional Farey sequence, and one hopes that techniques to study roots of congruences will continue to develop.
In this paper, we introduce the concept of block sub-additive potential. The topological and measure-theoretic pressures are then defined for the space of average pseudo-orbits relative to any block sub-additive potential and any open cover of a given compact metric space. A local variational principle connecting these pressures is established, and it is further proven that they are equivalent to the corresponding topological and measure-theoretic pressure (in the ergodic case), respectively, defined for the induced sub-additive potential and the specified open cover. Additionally, the global versions of these concepts are also investigated, and a result that bridges the global and local perspectives is presented.
Romain Aimino, Théophile Caby, Jorge Milhazes Freitas, Duarte Sá Pinho
math.DSmath.PR37A2537A5037B2037C3060G70
We establish Extreme Value Distributions for the closest encounter between trajectories generated by different maps defined in the same reference phase space. For a class of strongly mixing maps, we show that the limit distribution depends on the length of the different trajectories and the co-dimension of the associated invariant measures. It is also modulated by an Extremal Index, that informs on the tendency of nearby points to diverge along with the evolution of their respective dynamics, serving as an indicator of their compatibility. We give a formula for this quantity for a class of chaotic maps of the interval and for the co-dimension in the case when the respective measures admit densities with isolated zeros and singularities. We present diverse examples of systems satisfying these assumptions and compute the different parameters modulating the limit distribution.
Lyapunov exponents are fundamental invariants in smooth ergodic theory describing the asymptotic infinitesimal behavior along typical orbits. This text aims to explain how and why to control Lyapunov exponents using entropy for smooth surface diffeomorphisms. It fits into the framework of our recent joint works with Sylvain CROVISIER and Omri SARIG. We will focus especially on the continuity property of exponents for measures near the maximal entropy measure, by presenting a simplified version of the original argument. Our exposition is geared towards advanced students and researchers in dynamics that are not necessarily familiar with smooth ergodic theory.
We present some developments in the study of chaotic dynamics following the solution of a conjecture of Newhouse on the measures maximizing the entropy of smooth surface diffeomorphisms. We focus on strong positive recurrence, a generalization of the classical Anosov-Smale theory of uniform hyperbolicity introduced in a joint work with Sylvain Crovisier and Omri Sarig. This new property is general enough to be satisfied by all smooth surface diffeomorphisms with positive entropy, yet it still ensures many quantitative properties such as exponential mixing or limit theorems for regular functions. We also present some open problems, including its abundance (or not) in higher dimensions.
We study the existence of stable invariant measures for operators and strongly continuous semigroups of operators on Banach spaces admitting either a dense bilateral backward orbit or a sufficiently rich family of eigenvectors. These invariant measures are realized as the distributions of stochastic integrals with respect to stable random measures. We also discuss invariant measures with other classes of distributions for such operators and semigroups.
arXiv:2503.06597v5·Updated Jun 28, 2026·Originally submitted Mar 9, 2025
This work highlights a peculiar phenomenon of interval exchange. Considering a real number beta less than -1, the negative beta-shift is coded if and only if its absolute value is greater than the golden ratio. We study an increasing sequence of algebraic integers with limit-1 and the absolute value of the first term equals to the golden ratio such that for a base x taken in the interval of consicutive terms of this sequence, the measure of the maximal entropy is carried by the image of a beta-shift, with the golden ratio les than the absolute value of beta, under the mapping of an injective substitution.
We prove that epsilon multiplicity can take transcendental values. The main structural result is a one-ideal formula for section rings: under natural positivity hypotheses, the epsilon multiplicity of an ideal generated in one degree is equal to an integral of a divisor-volume function. This formula transports an asymptotic colength invariant of ideals to the geometry and arithmetic of divisor volumes. To produce a transcendental value, we combine the formula with a shifted projective-bundle construction inspired by Bornträger and Nickel. The shift places the construction in the positivity range required by the one-ideal formula while preserving the underlying disk geometry of the volume computation. Reversing the order of integration reduces the resulting integral to three integrals of rational functions. Their arctangent terms cancel exactly, whereas the remaining real logarithms form an explicit algebraic linear combination whose value is positive. Baker's theorem then implies transcendence. Consequently, there exists a homogeneous ideal in a normal standard graded domain whose epsilon multiplicity is transcendental.
Let k≥2 be fixed. We study the distribution modulo one of the nk sums a1+⋯+ak,1≤a1,…,ak≤n. counted with multiplicity. For S(h,n)=n/2≤a≤n∑e(ha),e(x)=exp(2πix), we prove second- and fourth-moment estimates matching the diagonal scale up to a factor nε. More precisely, H/2≤h≤H∑∣S(h,n)∣2≪ε,δHn1+ε uniformly for H≥n1/2+δ, and H/2≤h≤H∑∣S(h,n)∣4≪ε,δHn2+ε uniformly for n1/2+δ≤H≤n2/3, where 0<δ<1/6 in the fourth-moment estimate. Combining the second-moment bound with pointwise exponential-sum estimates and the Erdős--Turán inequality, we obtain Dk(n)≤n−ρk+o(1),ρk=26k+11671k+26, as n→∞, where Dk(n) denotes the discrepancy with respect to arbitrary subintervals of [0,1).
Let (an)n≥1⊂N be a lacunary sequence, an+1≥qan for q>1. For x∈T, we study the maximal empty circular gap GN(x) of the finite orbit {a1x,…,aNx}. We prove that, for Lebesgue-almost every x, 21≤N→∞liminflogNNGN(x)≤N→∞limsuplogNNGN(x)≤q−1q+1. If, in addition, an∣an+1 for every n, then this can be improved to N→∞limlogNNGN(x)=1 for Lebesgue-almost every x.
This paper is devoted to the study of Bowen's dimensional entropy on subsets for actions of amenable groups. We prove three main results. (1) First, topological conditional entropy is characterized by the dimensional entropy of stable sets (Theorem 1.1), answering a question of Dou, Wang and the second author of the present paper raised in [Fund. Math., 2025]. We remark that our Theorem 1.1 is the first characterization of topological conditional entropy via Bowen's dimensional entropy of stable sets even for Z-actions. (2) Second, we establish a dimensional entropy inequality for factor maps (Theorem 1.2). It relates dimensional entropy of a set to that of its image and topological entropy of fibres, and may be viewed as the dimensional-entropy counterpart of the factor-map inequality for packing topological entropy due to Dou, Zheng, and Zhou proved as Theorem 1.4 in [Ergodic Theory Dynam. Systems, 2023]. (3) Third, the relative topological entropy of a factor map is determined by the dimensional entropy of the fibres (Theorem 1.3). Notably, our proof of this formula (Theorem 1.3) is purely topological, in contrast to the recent measure-theoretic approach of Dou, Wang and Zhou based on relative Shannon--McMillan--Breiman theorems. These results (Theorem 1.2 and 1.3) not only generalize the work of Oprocha and the second author of the present paper [Nonlinearity, 2011] from single transformations to amenable group actions, but also provide a purely topological and self-contained proof of a fibre entropy characterization recently obtained through measure-theoretic arguments.
arXiv:2606.05973v2·Updated Jun 26, 2026·Originally submitted Jun 4, 2026
A metric graph with a mechanical Hamiltonian on each edge does not, by itself, define a deterministic classical motion through a branching vertex: conservation of energy fixes only the outgoing speed, not the outgoing edge-end. We study the deterministic problem obtained after this missing vertex datum is supplied. On each edge e, with coordinate q∈[0,ℓe], the Hamiltonian is He(q,p)=p2/2+Ve(q), where Vis continuous on the graph and C2 on every edge. At each vertex we prescribe an energy-preserving Borel isomorphism from incoming to outgoing nonzero boundary covectors. The resulting phase space is the measurable quotient that identifies each incoming boundary covector with its prescribed outgoing one. After excluding the finitely many energy levels V(v), the edgewise Hamilton equations and the vertex laws concatenate to a global one-parameter group of bimeasurable transformations. The group preserves energy and the quotient measure induced by the edgewise Liouville measures dqdp. The proof uses no smooth symplectic structure on the quotient; the invariance follows from ordinary edgewise Liouville invariance, a uniform no-Zeno estimate on compact regular energy windows, and preservation of the transverse Liouville flux rdr by the speedwise vertex permutations. If the vertex laws are compatible with momentum reversal, then the quotient dynamics is reversible. On regular energy surfaces satisfying the usual regular-value condition, the induced time-parametrization measure is invariant as well.
The competitive equilibrium of general equilibrium theory exists as a fixed point and is, by the theorys own results on aggregate excess demand, in general silent on whether that fixed point is unique, stable, or attained. This paper takes the economy to be not a configuration to be solved for but a process to be recovered, an asymptotically mean stationary information source carrying a partially identified operator of statistical dependence, populated by agents that are finite-capacity information channels. Within this adaptive order the competitive, rational expectations equilibrium is recovered exactly, as a joint limit taken along an explicit scaling path. Three parameter limits and two fixed-point conditions deliver it, the entropy rate falls to zero, agent channel capacity diverges, selection intensity grows infinitely sharp, adaptive learning reaches its expectationally stable rest point, and the recovered structure ceases to coevolve. At that corner the limiting object satisfies the axioms of the canon and its rest state is a Walrasian equilibrium, away from it the adaptive economy is a strict generalisation, carrying a positive entropy rate and a recovered dependence structure that the equilibrium primitive cannot express. We give the nesting as a theorem, establish the result by result correspondence with existence, with the Sonnenschein Mantel Debreu indeterminacy, and with the regular economies recovery, and characterise exactly what the equilibrium limit erases.
Andrzej Dąbrowski, Salah Eddine Rihane, Gökhan Soydan, Paul M. Voutier
math.NT11D4111D5911J8611Y50
Let Ps(n) denote the n-th s-gonal number. Consider the Diophantine equation Ps(n)=tm for integers n,s,t and m>2. All solutions to this equation are known for m>2 and s∈{3,5,6,8,10,20}. Here we extend these results to the cases s=2k+4 (where k=4,6 or 5≤k≤97 is a prime number) and s=k+4 (where k=9,15 or 3≤k≤97 is a prime number). The proofs of our results use the modular and hypergeometric methods, linear forms in logarithms and extensive calculations. We were unable to completely solve the above Diophantine equations, but we expect (based on GRH and the weak effective abc conjecture) that there will be no additional solutions beyond those explicitly shown in Theorems~1, 2 and 3.
arXiv:2507.23175v2·Updated Jun 26, 2026·Originally submitted Jul 31, 2025
Jalali and Poor introduced an asymptotic framework for compressed sensing of stochastic processes, demonstrating that any rate strictly greater than the mean information dimension serves as an upper bound on the number of random linear measurements required for (universal) almost lossless recovery of ψ∗-mixing processes, as measured in the normalized L2 norm. In this work, we show that if the normalized number of random linear measurements is strictly less than the mean information dimension, then almost lossless recovery of a ψ∗-mixing process is impossible by any sequence of decompressors. This establishes the mean information dimension as the fundamental limit for compressed sensing in this setting (and, in fact, the precise threshold for the problem). To this end, we introduce a new quantity, related to techniques from geometric measure theory: the correlation dimension rate, which is shown to be a lower bound for compressed sensing of arbitrary stationary stochastic processes.
arXiv:2605.15580v2·Updated Jun 26, 2026·Originally submitted May 15, 2026
We investigate actions of locally compact Abelian (LCA) groups on the torus Tn, motivated by their close connection with Diophantine approximation. While Kronecker's theorem yields a classical density result, we prove a stronger equidistribution theorem of Weyl type: every such action admits a decomposition into uniquely ergodic subsystems. The proof of this result is based on a characterization of unique ergodicity for actions of amenable groups on compact metric spaces. As consequences, we establish several foundational results for LCA groups, including the Bohr orthogonality of characters along arbitrary Folner sequences, a Bohr mean formula for almost periodic functions, and a Wiener-type theorem on LCA groups characterizing the discrete part of a Borel probability measure through its Fourier transform. An application to numerical analysis is also discussed.
The natural extension of the triangle map (a type of multi-dimensional continued fraction algorithm) is completely described in all possible dimensions. The motivation and inspiration for this natural extension stems from the triangle map's recent link to the classical study of integer partitions. Inspired by Young conjugation for integer partitions, we show that the natural extension has an internal symmetry and allows the natural extension to be subdivided into four natural subdomains. This appears to be new even for the classical case of the natural extension for continued fractions, namely for both the classical Gauss map and the classical Farey map.
In this paper we count the number of common values shared by two linear recurrence sequences, whose characteristic polynomials are a generalized Ankeny-Brauer-Chowla polynomial and its reciprocal. More precisely, we show that these sequences have at most two sufficiently large common values. Our proof combines Baker's theory of linear forms in logarithms of algebraic numbers with techniques from function field theory and from Galois theory.
Recent numerical computations and stochastic modeling by Brevitt and Klages suggest that introducing a hole in a Pomeau--Manneville map can suppress survivor-conditioned Lyapunov stretching. We prove a deterministic renewal theorem which explains this phenomenon and its observable-level generalizations. For an open intermittent map induced on a base away from the neutral fixed point, we describe the asymptotic distribution of the number of completed survivor returns to the base, conditioned on survival up to time t. The limiting law is expressed in terms of the killed induced transfer operator; for the conditionally invariant density of the killed induced system it is geometric. We then prove two reward results for additive observables. A reward domination theorem gives bounded survivor-conditioned Birkhoff sums, while a stronger final-tail asymptotic gives convergence to a finite limit. For generalized Pomeau--Manneville maps, bounded observables satisfying ∣ψ(x)∣≤Cxγ near the neutral fixed point and a mild variation condition satisfy the domination hypotheses. When the neutral branch and final tails satisfy the corresponding regularity assumptions, asymptotically regular observables satisfy the convergence hypotheses. In particular, ψ=log∣f′∣ gives bounded survivor-conditioned Lyapunov stretching for the generalized class; under these additional regularity assumptions, it converges. Under an additional entropy-domination assumption, we also derive a zero entropy-rate consequence for survivor return-length names and record the complementary linear growth of stretching when the hole contains a neighborhood of the neutral fixed point.
arXiv:2602.08196v3·Updated Jun 25, 2026·Originally submitted Feb 9, 2026
We study Schrödinger operators on quantum graphs where the number of edges between points is determined by orbits of a "shift of finite type". We prove Anderson localization for these systems.
We adapt ideas of Kim and Roush [15], originally developed in the study of automorphisms of sofic subshifts, to obtain sufficient conditions under which a subshift has a huge automorphism group. We apply this approach to non-sofic subshifts defined by sets of multiples. In particular, we establish a dichotomy for the B-admissible subshift: its automorphism group is either trivial or contains an embedded copy of the automorphism group of the full shift {0,1}Z. In the latter case, we say that the automorphism group is huge. We further show that the automorphism group of the hereditary closure of the B-free subshift is huge whenever B⊂N is infinite and contains no infinite pairwise coprime subset.
arXiv:2603.14116v2·Updated Jun 24, 2026·Originally submitted Mar 14, 2026
We prove, in particular, the well--known Zaremba conjecture from the theory of continued fractions for any prime denominator. More precisely, we show, firstly, that under some mild conditions, for any sufficiently large q, there exists a coprime to q such that all partial quotients of a/q are bounded by O(logq), and, moreover we find asymptotically tight lower bound for the number of such a. Secondly, we obtain a good lower bound for the number a such that the sum of all partial quotients of a/q is bounded by O(logq⋅loglogq). This, accordingly, improves on some results of Korobov and Larcher. Finally, we show that for all sufficiently large M there are Ω(q1−O(1/M)) numbers a coprime to q such that all partial quotients of a/q are bounded by M.
Let C be the middle-third Cantor set and μ be the Cantor-Lebesgue measure on C. A conjecture of Velani states that μ(W2(τ))=0 if τ>1 and μ(W2(τ))=1 if 0<τ≤1, where W2(τ)={x∈[0,1]:∥2nx∥<n−τforinfinitelymanyn∈N}. We prove that the conjecture holds for τ>γ1−3−γ1−γ(≈1.429) and 0<τ<12γ(≈0.052), where γ=log3log2 is the Hausdorff dimension of C. This improves the known results on both the null part (τ>γ1−γ(2−γ)0.078(1−γ)≈1.552, due to Allen, Baker, Chow, and Yu (2023)) and the full measure part (0<τ≤0.01, due to Baker (2025)). Our key innovation is to establish the estimate n=1∑N∣μ(h2n)∣2≪N1−γ and its consequences: n=1∑N∣μ(h2n)∣≪N1−2γ,n=1∑Nn−σ∣μ(h2n)∣≪σN1−2γ−σ, where 0<σ<1−2γ, and all estimates are uniform in h∈Z∖{0}. For the full measure part, our approach also generalizes to self-similar measures on a class of missing-digit sets.
arXiv:2205.12137v3·Updated Jun 23, 2026·Originally submitted May 24, 2022
We build quantitative measure subgroup couplings from a Brieussel-Zheng diagonal product to a lamplighter group. We use them to answer the inverse problem of the quantification; namely, find a group admitting a measure subgroup coupling with a prescribed group with prescribed quantification, in the case of the lamplighter group.
arXiv:2605.12360v3·Updated Jun 23, 2026·Originally submitted May 12, 2026
We study the two ℓ2-Dirichlet structures on a countable group G arising from the left and right regular actions on RG. Although the two regular representations are unitarily equivalent, their ℓ2-Dirichlet subspaces of RG need not coincide. Our main result gives a complete classification of this asymmetry for countable amenable groups: D2(G,λ)=D2(G,ρ)⟺G is an FC-group. The proof is based on a skewed Følner-geometric mechanism, called a left scheme, combining summability of left boundaries with displacement under a right translation. We develop this mechanism generally, and demonstrate it concretely in the Heisenberg group and amenable wreath products over Z. We also show that this mechanism has a dynamical counterpart in the theory of nonsingular Bernoulli shifts: every countable amenable group that is not an FC-group admits Bernoulli schemes whose left shift is nonsingular, conservative and weakly mixing, whereas the right shift by some element is singular.
arXiv:2605.17529v2·Updated Jun 22, 2026·Originally submitted May 17, 2026
We construct Bohr obstructions to multiple recurrence along rounded Hardy-field sequences, showing that the real derivative-span criterion of Bergelson, Moreira, and Richter is essentially sharp and answering two of their questions. For E⊆N and u:N→Z, set Ru(E):={n∈N:E∩(E−u(n))=∅}. We prove that, if f1,…,fk are functions of polynomial growth from a Hardy field and some real linear combination of f1,…,fk and their derivatives has a nonzero finite limit, then there exist M∈N and a basic Bohr set E⊆N such that ⋂i=1kR[Mfi](E) is not thick. In particular, for some Bohr set E, the set R[t3/2](E)∩R[2t3/2+t](E) is piecewise syndetic but not thick. We also prove that, if for some λ1,…,λk∈R we have x≥1infi∑λifi(x)T>21i∑∣λi∣, then ⋂iR[fi](E)=∅ for some basic Bohr set E. More generally, our results apply with [⋅] replaced by any rounding function ρ:R→Z satisfying supx∈R∣ρ(x)−x∣<∞.
For any infinite one-relator group Γ=⟨S∣wm⟩, we prove that cost(Γ)=∣S∣−m1. For such groups, this gives β1(2)(Γ)=cost(Γ)−1, answering a special case of Gaboriau's question on the relationship between cost and first ℓ2-Betti number.
We prove the positivity and continuity of the Lyapunov exponent for one-dimensional discrete Schrödinger operators with multi-frequency skew-shift potentials. For the operator Hλ,ω=Δ+λv(Tωn(x,y)) on ℓ2(Z), where Tω is a skew-shift on Td×Td(d≥1) and v is a non-constant real-analytic function on T2d, we establish that for Diophantine frequency vectors ω and large coupling λ≫1, the Lyapunov exponent satisfies L(λ,E)≥clogλ>0 uniformly in E (with c>0), and is log-Hölder continuous in E. This work extends the known results of Lyapunov exponents--previously developed for one-frequency or simpler quasi-periodic models--to the genuinely multi-frequency skew-shift setting.
arXiv:2408.06200v3·Updated Jun 22, 2026·Originally submitted Aug 12, 2024
For a norm F on R2, we consider the set of F-Dirichlet improvable numbers DIF. In the most important case of F being an Lp-norm with p=∞, which is a supremum norm, it is well-known that DIF=BA∪Q, where BA is a set of badly approximable numbers. It is also known that BA and each DIF are of measure zero and of full Hausdorff dimension. Using classification of critical lattices for unit balls in Lp, we provide a complete and effective characterization of DIp:=DIF[p] in terms of the occurrence of patterns in regular continued fraction expansions, where F[p] is an Lp-norm with p∈[1,∞). This yields several corollaries. In particular, we resolve two open questions by Kleinbock and Rao by showing that the set DIp∖BA is of full Hausdorff dimension, as well as proving some results about the size of the difference DIp1∖DIp2. To be precise, we show that the set difference of Dirichlet improvable numbers in Euclidean norm (p=2) minus Dirichlet improvable numbers in taxicab norm (p=1) and vice versa, that is DI2∖DI1 and DI1∖DI2, are of full Hausdorff dimension. We also find all values of p, for which the set DIpc∩BA has full Hausdorff dimension. Finally, our characterization result implies that the number e satisfies e∈DIp if and only if p∈(1,2)∪(p0,∞) for some special constant p0≈2.57.
In this note we solve a problem posed by Lagarias and Shallit concerning Lagrange constants under linear fractional transformations Mx=cx+dax+b. For an integer matrix M with nonzero determinant and relatively prime entries, define the ratio spectrum V(M)={k(x)k(Mx):x∈Bad}, where k(x) denotes the Lagrange constant of the irrational number x and Bad is the set of badly approximable numbers. Lagarias and Shallit proved that V(M)⊆[∣detM∣1,∣detM∣], and asked for the determination of V(M). We prove that V(M)=[∣detM∣1,∣detM∣].
In a recent paper, Chatterjee, the author and Mohan posed the problem of determining all solutions of the Diophantine equation Fn=Φ(m), where Fn is the n-th Fibonacci number and Φ(m) counts the number of nonempty sets A⊆{1,2,…,m} for which gcd(A) is relatively prime to m. In this paper, we prove that the Diophantine equation has the only solutions (n,m)=(1,1),(2,1),(3,2). The main tools used in this paper are lower bounds for linear forms in logarithms due to Matveev and Dujella-Peth{ő} version of the Baker-Davenport reduction method in diophantine approximation.
arXiv:2606.15843v2·Updated Jun 18, 2026·Originally submitted Jun 14, 2026
We study dynamical orthogonal (DO) approximations of stochastic differential equations and investigate their long-time behaviour. The DO formulation represents the solution by a low-rank decomposition and leads to a coupled system consisting of an evolution equation on the Stiefel manifold and a reduced stochastic process. We establish the well-posedness of the strong DO system and derive quantitative error estimates between the original stochastic differential equation and its low-rank approximation in the Wasserstein distance. Our main contribution is the analysis of invariant probability measures for the DO dynamics. Under suitable dissipativity, Lipschitz continuity, and non-degeneracy assumptions on the coefficients, we prove the existence of an invariant probability measure for the strong DO system. The proof combines uniform moment estimates, a Krylov--Bogoliubov argument for an associated frozen system, and a Kakutani-Fan-Glicksberg fixed-point theorem to recover the self-consistent dynamics. We further show that the induced low-rank process admits an invariant probability measure and discuss the structure of invariant measures through several illustrative examples. These results provide a rigorous foundation for the use of dynamical low-rank approximations in the approximation of long-time statistical properties of stochastic dynamical systems.
arXiv:2510.19441v2·Updated Jun 17, 2026·Originally submitted Oct 22, 2025
The modeling of diffusion processes on graphs is the basis for many network science and machine learning approaches. Entropic measures of network-based diffusion have recently been employed to investigate the reversibility of these processes and the diversity of the modeled systems. While results about their steady state are well-known, very few exact results about their finite-time evolution exist. Here, we introduce the conditional entropy of heat diffusion in graphs, and outline a mathematical framework that contextualizes diffusion and conditional entropy within the theories of continuous-time Markov chains and information theory. In particular, we highlight that this entropic measure satisfies an information-theoretical version of the second law of thermodynamics, thereby providing a parallelism between diffusion dynamics on networks and their physical counterparts. Furthermore, we obtain explicit results for its evolution on complete, path, and circulant graphs, as well as a mean-field approximation for Erdös-Rényi graphs. We also obtain asymptotic results for general networks and provide bounds for the evolution of conditional entropy. Finally, we experimentally demonstrate several properties of conditional entropy for diffusion over random graphs, such as the Watts-Strogatz model.
arXiv:2302.14839v2·Updated Jun 17, 2026·Originally submitted Feb 28, 2023
We study the behavior of the pressure function for Hölder continuous potentials on mixing subshifts of finite type. The classical theory of thermodynamic formalism shows that such pressure functions are convex, analytic and have slant asymptotes. We provide a sharp exponential lower bound on how fast the pressure function approaches its asymptotes. As a counterpart, we also show that there is no corresponding upper bound by exhibiting systems for which the convergence is arbitrarily slow. However, we prove that the exponential upper bound still holds for a generic Hölder potential. In addition, we determine that the pressure function satisfies a coarse uniform convexity property. Asymptotic bounds and quantitative convexity estimates are the first additional general properties of the pressure function obtained in the settings of Bowen and Ruelle since their groundbreaking work more than 40 years ago.
We introduce dynamical blanket times, which quantify how quickly the empirical distribution along a typical orbit approximates the invariant measure. These can be viewed as a measure-theoretic analogue of the previously introduced dynamical cover time, which measures how quickly an orbit becomes dense in the space. Motivated by analogous comparability questions for random walks on graphs, we investigate how much longer it takes for a dynamical system to ``blanket'' than to ``cover''. For finite-branch, uniformly expanding interval maps, we obtain upper bounds on the expected blanket time in terms of the spatial scale and the precision of approximation. In the special case where the invariant measure is absolutely continuous with respect to Lebesgue, this yields comparability between the expected blanket and cover times, uniformly across all sufficiently small scales. Our approach combines two main ingredients. First, we establish large deviation estimates for hitting times which are uniform over both the target location and the spatial scale. Second, using methods from multifractal analysis, we construct a finite discretisation of the invariant measure which reduces the problem to a suitably controlled discrete model.
arXiv:2603.16794v3·Updated Jun 17, 2026·Originally submitted Mar 17, 2026
We prove that for any real number ξ=0 and any coprime integers p>q≥1 such that ξ is irrational or q>1, the image in R/Z of the sequence (ξ(−p/q)n)n≥0 is not contained in any interval of length less than (1+q/p−q2/p2)/p.
Let Γ be a geometrically finite thin subgroup of an arithmetic lattice Γ0<G:=SO(n,1) and consider the congruence covers of Γ\G. In the breakthrough work of Bourgain-Gamburd-Sarnak, the expansion machinery was used to establish a uniform spectral gap in the setting (G,Γ0)=(SL2(R),SL2(Z)) when the critical exponent satisfies δΓ>21. The main applications are affine sieve for Γ-orbits and uniform resonance-free half-planes for the resolvent of the Laplacian. These results were generalized in subsequent works by Mohammadi-Oh, Oh-Winter, the author, and Edwards-Oh. Yet, the region δΓ∈(21,n−2] for n≥3 remains to be treated when there are cusps. The purpose of this paper is to fill in this gap in the literature. The difficulty lies in working with a countably infinite coding due to the presence of cusps. In particular, we incorporate new tools to prove the Zariski density and full trace field properties of the return trajectory subgroups.
arXiv:2408.11274v3·Updated Jun 17, 2026·Originally submitted Aug 21, 2024
Let Γ be a Zariski dense Anosov subgroup of a connected semisimple real algebraic group -- these are higher rank analogues of convex cocompact subgroups. Let us measure the Jordan projections with any linear form which is positive on the limit cone of Γ. We prove a corresponding counting theorem with a power saving error term for the conjugacy classes of loxodromic elements in Γ. The proof is based on interpreting the Jordan projections as periods of a natural flow associated to Γ and proving exponential mixing. We also prove the existence of a spectral gap for the Selberg zeta function.
In this paper, we show that there are 57 nonnegative integer solutions (a,b,x) to the inequality 1≤∣x2−2a⋅3b∣<3max{a,b} and we list them explicitly. The inequality is converted into a statement about how closely x/q approximates irrational number d for d∈{2,3,6}, where q is an integer which is 3-smooth, after which Worley's theorem on rational approximations via continued fractions is applied to parametrise the solutions and a p-adic lower bound for a linear form in logarithms due to Bugeaud and Laurent is applied to find a rather large bound on max{a,b}. We finish with an application of the LLL algorithm to reduce this bound.
We investigate the asymptotic behavior of the Diophantine errors δBn−An, where δ=−eEi(−1) denotes the Euler-Gompertz constant and Bn and An are the nth Bell and Gould numbers, respectively. These errors have exponential generating function gδ(z)=exp(ez−1)(δ−∫0zexp(1−et)dt),z∈C, and it is known that δBn−An=O(Bnexp(−cn/(ln(n))2)) for some c∈R>0, implying limn→∞(An/Bn)=δ. In the present work, we prove that δBn−An oscillates without bound as n→∞; that is, both limsupn→∞(δBn−An)=∞ and liminfn→∞(δBn−An)=−∞.
arXiv:2109.14179v3·Updated Jun 16, 2026·Originally submitted Sep 29, 2021
We show that if Z3 can be tiled by translated copies of a set F⊆Z3 of cardinality the square of a prime then there is a weakly periodic F-tiling of Z3, that is, there is a tiling T of Z3 by translates of F such that T can be partitioned into finitely many 1-periodic sets.
arXiv:2603.20483v2·Updated Jun 16, 2026·Originally submitted Mar 20, 2026
Ludovick Bouthat, Philippe-André Luneau, Philippe Petitclerc
math.NAmath.PR65H0560G1047N4037H9937A50
We propose a generalized version of the bisection method where the cutting point between the two subintervals is chosen at random following an arbitrary distribution. We compute expected convergence rates with respect to any arbitrary a priori distribution for the position of the root in the initial interval and proved that it depends only on the the expectation E[c(1−c)] of the cut c. We also provide a generalization of the method for K random cuts and study its convergence properties. Most probabilistic derivations are kept fairly simple for the ease of understanding of a larger audience. Our theoretical results are then validated numerically using statistical simulation.
The irrationality of e+pi remains open, despite the separate transcendence of e and pi. This paper studies the problem from the viewpoint of finite irrationality certificates and gives a bounded no-go audit for low-complexity Apéry-type proof mechanisms. First, we prove exact equivalences between the hypothesis e+pi in Q and eventual factorial-arithmetic phenomena: a ceiling recurrence, a factorial-Cantor digit condition, and a divisibility criterion. These criteria identify what rationality would force, while showing why tail conditions are not finite obstructions. Second, we formulate an Apéry-type certificate framework based on integer linear forms L_n = A_n(e+pi)+B_n with A_n,B_n in Z, L_n nonzero, and |L_n| tending to zero. A mixed integration-by-parts identity produces such forms from integer polynomials. We then audit several low-complexity constructions, including mixed Padé approximation, crossed separate approximations to e and pi, simple J-fractions, holonomic ansatzes, Rodrigues-type families, and an integer kernel-lattice search. The main contribution is a rigid boundary probe: no-go filters marking a tested zone where analytic smallness is destroyed by denominator clearing, coefficient growth, primitive reduction, or continued-fraction shadows. In the final kernel-lattice audit, 145 raw candidates reduce to 133 primitive records; the best signals are dominated by continued-fraction shadows, while non-CF candidates do not form a degree-continuing family. Thus, within the tested low-complexity families, no non-circular Apéry-type mechanism for e+pi is found.
We prove a quantitative version of Oppenheim's conjecture in the function field setting. In order to do so, we compute the higher moments of the Siegel transform. In particular, we find an optimal bound on the variance of the number of lattice points in a set. Moreover, we compute the exact variance of the number of lattice points in a ball, which is of independent interest.
Consider the following plaquette model from statistical physics: a lamp lies at every vertex of the triangular lattice and a switch lies at every even vertex of the (bipartite) dual hexagonal lattice. Each switch toggles the three lamps on its face. The energy of a configuration is the number of ON lamps. For the Glauber dynamics associated with the Gibbs measure defined by this Hamiltonian at any inverse temperature β>0, we show that, in any dimension d≥2, the infinite volume relaxation time satisfies eβ2/C/C≤Trel≤CeeCβ for some C>0. Our result entails that the Gibbs measure is unique. The eβ2 scaling was conjectured by Newman and Moore in 1999 and matches the behaviour of supercritical rooted kinetically constrained models such as the East model, thus recovering fragile glass phenomenology in the absence of kinetic constraints. More precisely, we show that, on a torus of side length 2k, when β→∞ and k/β→0, we have Trel=e2βk(1+o(1)). Quite surprisingly, however, we also prove that, on non-periodic finite domains of size n≤eβ/C for large C>0, we have the much larger asymptotics lnTrel=βnΘ(1). The main ingredients of the proofs are new results in extremal and enumerative combinatorics and rely on renormalisation ideas for the dynamics and its groundstates also known as the Ledrappier subshift. We note consequences of our results to geometric group theory (more precisely to the complexity of the word problem for the Baumslag finitely presented group) and to ergodic theory.
arXiv:2510.12596v3·Updated Jun 14, 2026·Originally submitted Oct 14, 2025
We establish distributional laws for Poincaré recurrence in measure-preserving systems (X,T,μ) satisfying an exponential multiple decorrelation condition and a short returns condition. When the measure is absolutely continuous, the sum ∑k=1n1B(x,rk)(Tkx)−μ(B(x,rk)) does not in general obey a CLT; instead, it converges to a non-standard distribution that is an average of Gaussian laws weighted by the density of μ. By considering a version of the sum where we appropriately rescale the radii of the balls, we recover the CLT. A key assumption in our recurrence theorems is that the corresponding hitting sums satisfy the CLT. We verify this assumption for Axiom A systems by establishing the stronger ASIP for the shrinking target problem, extending Haydn, Nicol, Török and Vaienti [Trans. Amer. Math. Soc. 2017] and related results. Systems for which our results apply include piecewise expanding systems on the interval, and Axiom A systems. The results highlight the difference between recurrence and hitting behaviour.
This paper is devoted to the study of uniformly distributed (u.d.) sequences on general non-compact topological spaces. Convergence-determining classes and the equivalent characterizations of u.d. sequences on T4 spaces are discussed. For Polish spaces endowed with regular probability measures, the existence of a countable convergence-determining class for u.d. sequences is proved by virtue of the tightness of Borel probability measures. It is shown that, under the condition that a countable convergence-determining class exists, both independent identically distributed (i.i.d.) randomly sampled sequences and orbit sequences of ergodic transformations are u.d.\ with full measure.
In this paper, we study the existence of invariant sublinear expectations of Markovian semigroups on sublinear expectation spaces. To achieve this, we establish a complete metric space of sublinear expectations, on which we extend Harris' method to the nonlinear setting on the convergence of sublinear semigroups. We then explore two cases of G-diffusions by studying the Lyapunov function and the local Doeblin condition. One is the G-Brownian motion on the unit circle which is the case studied in Feng and Zhao, but with the new method. Another is the multidimensional G-SDEs on the whole space Rd. We establish, for the first time in the literature, the existence of the invariant sublinear expectation for G-SDEs under the non-degenerate and weakly dissipative assumption. For this, we prove that for a class of G-SDEs, the G-expectation can be represented as the supremum of the semigroup of a family of SDEs, of which the regularity is obtained by considering the Bismut-Elworthy-Li formula and the Denis-Hu-Peng representation for the distribution of G-Brownian motions.
In this paper, we extend Tsukamoto's recent localization formula for metric mean dimension to actions of countable discrete amenable groups, which previously applied only to Rk- and Zk-actions -- by proving that the global metric mean dimension is characterized by the asymptotic entropy of pointwise ε-stable sets (Theorem 2.3). To achieve this generalization, we introduce equivalent definitions of the invariant using topological entropy, packing topological entropy, and Bowen's dimensional entropy, respectively. A key technical contribution is our replacement of tiling arguments with Lindenstrauss's combinatorial covering lemma, which enables us to handle the general structure of amenable groups. Furthermore, we resolve all three questions regarding uniformity raised in Section 6 of a recent paper by Yang, Chen, and Zhou by constructing counterexamples (Theorem 2.6 and Proposition 5.8), which demonstrates that the supremum and limit superior in the localization formula cannot generally be interchanged, thereby highlighting the heterogeneous nature of the convergence. These results clarify the uniformity issue and offer insights into the link between local dynamics and global invariants, while our equivalent definitions provide flexible tools for computing metric mean dimension in concrete settings.
Given an ordered tuple v=(v0,…,vd) of vectors in Rd, let Av=[v1−v0⋯vd−v0] be its edge matrix. We prove that, in every finite colouring of Rd, one colour class realizes every prescribed value of the higher characteristic coefficients (c2(Av),…,cd(Av)). This extends Graham's theorem on volumes, which corresponds to the last coefficient cd(Av)=det(Av). We also prove a discrete analogue: if E⊆Zd has positive upper Banach density, then, for some q≥1, the set of coefficient tuples realized by ordered tuples in E contains q2Z×q3Z×⋯×qdZ. Finally, we show that the ordinary trace c1(Av) cannot be added to these conclusions. The proof combines a quantitative directional expansion result for ergodic actions of free abelian groups with a trace calculation for a family of model edge matrices.
arXiv:2405.17045v2·Updated Jun 9, 2026·Originally submitted May 27, 2024
We investigate Ruelle-Pollicott resonances of smooth Anosov diffeomorphisms, acting on manifolds of every dimension, with respect to the measure of maximal entropy. We highlight a profound connection between resonances and eigenvalues of the action induced by the dynamics on de Rham cohomology. In particular, resonances appear as eigenvalues of a quasi-compact transfer operator acting on suitable anisotropic spaces of currents. After defining the anisotropic Banach spaces, we introduce the anisotropic de Rham cohomology and we show that it is isomorphic to the standard de Rham cohomology. The relation between resonances and cohomological eigenvalues is deduced from a comparison of spectra. We finally exploit these results to get information about the Ruelle-Pollicott asymptotics of the correlation function and to establish a cohomological bound for the speed of mixing of Anosov diffeomorphisms.
Let λ=λ(β,σ,a,b) denote the top Lyapunov exponent for the linearization along trajectories of the noisy damped non-linear oscillator x¨+βx˙+ax+bx3=σW˙t, where a, b and β are all positive and σ=0. In 2004 Arnold, Imkeller and Sri Namachchivaya stated without proof that λ(ε2β,εσ,a,b)∼λε2/3 as ε→0 with λ>0. This paper contains a proof of this assertion.
We exhibit a family of smooth incompressible velocity fields on the two-dimensional unit sphere such that the time evolution of any mean-free initial data passively advected by any of them is mixed exponentially fast. In the presence of molecular diffusivity, we show that the solution to the associated advection-diffusion equation experiences enhanced dissipation with optimal decay rates. Each member of this family is an alternating combination of two Rossby-Haurwitz flows with random amplitudes and constitutes a spherical analogue to the sine shear-alternating example of Pierrehumbert.
To construct obstructions to the stability of rational maps with non-summable critical points in their Julia sets, we introduce automorphic measures with complex eigenvalues for rational maps on the Riemann sphere. In particular, these measures extend the classical notions of quasi-invariant and conformal measures by allowing the respective Radon--Nikodym derivative to be complex-valued and proportional to a multiplicative cocycle. j(s,t)(R)=∣R′∣s(R′∣R′∣)t, which plays the role of a generalized automorphy factor in the sense of group actions. The existence of such measures reveals a close connection between geometric and dynamical properties of rational maps. We show that the existence of certain automorphic measures, particularly unimodular measures and their associated vector fields, implies instability of the corresponding rational map. Specifically, for a weakly dissipative rational map admitting a (−1,1)-unimodular measure, there exists an integer q≥1 such that the map is q-unstable. This result generalizes earlier instability criteria involving pseudoconformal measures and connects the presence of such measures to the failure of structural stability. Furthermore, we establish ergodic and combinatorial conditions ensuring the existence of unimodular measures or vector fields -- most notably, through bounded recurrence, bounded velocity of arguments, and relations with the Milnor--Thurston kneading theory. These criteria provide a unified framework linking automorphic measures, Lyapunov spectra, and the geometric deformation spaces of rational maps.
We prove local centralizer rigidity results for elements of Weyl chamber flows and twisted Weyl chamber flows on compact homogeneous spaces. For generic elements in the twisted setting, sufficiently large dimension of the centralizer forces smooth conjugacy to an algebraic model. For many non-generic elements, we prove analogous rigidity under a virtual centralizer-isomorphism hypothesis.
We prove that the one-dimensional Dirichlet spectrum with respect to approximation in L1 norm D[1] satisfies D[1]=[21,1]. This is equivalent to the fact that the Minkowski spectrum M, associated with the Minkowski diagonal continued fraction, satisfies M=[41,21]. Further, we show that level sets Θm={α∈(0,1)∖Q:m(α)=m}, where m(α) is the Minkowski constant of α, have Hausdorff dimension strictly greater than 1/2 for any m∈(1/4,1/2], while dimHΘ1/4=21.
Jason Bell, Colin Ingalls, Rahim Moosa, Matthew Satriano
math.AGmath.LOmath.RA12H0503C9832M2513N1511J95
A differential analogue of the conjecture of Reichstein, Rogalski, and Zhang in algebraic dynamics is here established: if X is a projective variety over an algebraically closed field of characteristic zero which admits a global algebraic vector field v:X→TX such that (X,v) has no proper invariant subvarieties then X is an abelian variety. Vector fields on abelian varieties with this property are also examined. Some of the analysis works in the more general context of D-varieties over differential fields: projective D-varieties without proper D-subvarieties are homogeneous. But the main theorem does not extend: an example of a D-variety structure on the projective line without proper D-subvarieties is given.
We study Birkhoff genericity for a simple uniformly expanding diagonal flow on SLn+1(R)/SLn+1(Z), with initial points restricted to affine subspaces of the expanding horospherical orbit through the identity coset. We prove that almost every point on such an affine subspace is Birkhoff generic, except possibly in two situations: either the defining matrix of the affine subspace has Diophantine exponent at least n, or the affine subspace is arbitrarily well approximable by affine subspaces of dimension (r−1) defined over a real number field of degree m≥2, with n+1=mr. As applications, we obtain Dirichlet non-improvability and logarithmic density results for almost every point on these affine subspaces.
We begin by presenting the mathematical rationale underlying classical deductive inference. We then introduce the foundational ideas of the Bayesian inference framework. Results lying at the interface of Statistics and Ergodic Theory are outlined, providing a theoretical framework applicable to the prediction and analysis of real-world phenomena from random data. This text is expository in nature - no new results are presented; rather, recently published results are described in a didactic manner. Throughout, we work with Hölder equilibrium measures, which encompass a substantially more general class of processes than i.i.d. ones.
We study the spectral properties of Laplacians on the Schreier graphs Γn of the basilica group, the iterated monodromy group of the polynomial z2−1, which is an important example in the theory of self-similar, amenable but not elementarily amenable, automaton groups. Building heavily on results by Brzoska, Jarvis, George, Rogers and Teplyaev about certain subgraphs of the basilica graphs, we develop a new recursive framework for computing the characteristic polynomials of these Laplacians. Our analysis reveals a simple underlying dynamical system and proves approximation results for the Kesten-von Neumann-Serre (KNS) spectral measure.
We study finite-sample change detection for one-dimensional noisy dynamical systems using partition-based empirical approximations of stationary behaviour. Given observations from an interval-valued process, we partition the state space, estimate a finite transition matrix from observed transitions between partition elements, and apply a small Doeblin-type regularisation to ensure a unique stationary distribution. From an initial reference segment, we compute a baseline empirical stationary distribution π0,ρ. For each later sliding window, we compute πt,ρ and define the score St=∥πt,ρ−π0,ρ∥1. Large values of St indicate a change in stationary behaviour relative to the baseline. The statistic detects changes in invariant density or stationary law, but not all possible changes in transition dynamics. Under explicit assumptions on empirical transition concentration, finite-state stationary distribution stability, partition approximation, regularisation bias, and noise stability, we derive a finite-sample bound for the empirical stationary density. The bound separates sampling error, regularisation bias, partition approximation error, and noise bias. We then obtain a single-window false-alarm guarantee and a sufficient detection condition when the invariant density changes by more than the estimation error. We illustrate the method on synthetic noisy beta-map change-point experiments.
We generalize Voronoi's theory of perfect quadratic forms to generalized copositive matrices over a full-dimensional closed convex cone K, by introducing a K-copositive minimum and perfect K-copositive matrices. We consider a key feature of a given cone, which we call Interior Ryshkov (IR) property. Under this property the classical theory and its applications generalize nicely and we prove that rationally generated cones possess this IR property. For contrast, we give a detailed example of a simple cone without the IR property, showing various differences to the classical case. Moreover, this example yields connections to questions of number theory, in particular to Diophantine approximation and the Pell Equation. Finally, as an application, we give inner and outer polyhedral approximations for the generalized completely positive cone and a method to find rational certificates for (non-)membership in this cone.
arXiv:2601.21954v2·Updated Jun 4, 2026·Originally submitted Jan 29, 2026
This paper establishes an asymptotic expansion for expanding sector average on compact quotients of real rank-one Lie groups, with SO(n,1) as a model case. Using tools from harmonic analysis and representation theory, the problem is reduced, via the action of the Casimir operator, to an ordinary differential equation(ODE) governing the evolution along the expanding A-direction. The asymptotic expansion is then derived from the analysis of this ODE.
We prove the finiteness of the set of j-invariants of Drinfeld modules of rank 2 over Fq[T] which are CM and S-units, for S the infinite set of primes with even degrees. The proof is based on the study of ordinary reduction and supersingular reduction of Drinfeld modules, and on the splitting behaviour of primes dividing the difference of two Drinfeld singular moduli. We also provide an algorithm to compute a polynomial with coefficients in Fq[T] and roots the j-invariants having CM by a given order, and use it to compute some explicit examples, providing for instance counterexamples to a conjecture of Dorman. For a maximal order O, we prove by a universality argument that our algorithm computes the Hilbert modular polynomial HO.
We establish Diophantine approximation results for real symmetric matrices by collections of linearly independent integer vectors. For X∈Symd(R), we prove a Dirichlet-type theorem guaranteeing the existence of integral Lagrangian frames (Q,P)∈Matd×2d(Z) that satisfy ∥QX+P∥op≤cd/N and ∥Q∥op≤N for any N≥1. Furthermore, we establish a Khintchine-type zero-one law, demonstrating that the size of the set of ψ-approximable symmetric matrices is determined by the convergence or divergence of the series ∑q≥1qς−1ψ(q)ς, where ς=d(d+1)/2. The proofs rely on the reduction theory of the Siegel upper half-space, dynamical formulation over the space of symplectic lattices, and an analysis of the Siegel transform adapted to count Lagrangian frames instead of single lattice points.
In this article, we propose an algorithmic approach to determine the integer points located near a transcendental curve. This approach is closely related to a celebrated work by Bombieri and Pila and to the so-called Coppersmith's method. We establish the underlying theoretical foundations, prove the algorithms, study their complexity and present practical experiments; we also compare our approach with previously existing ones. From a practical point of view, we focus on an instance of our general problem, called the Table Maker's Dilemma, whose solving makes it possible to evaluate a given function with correct rounding. Our experiments show a significant speedup. In particular, our results show that the development of a correctly rounded mathematical library for the binary128 format is now possible at a much smaller cost than with previously existing approaches.
We prove a No-Core Principle for stationary actions of countable groups. Namely, if a Borel set intersects almost every orbit in finitely many points and has positive measure, then it is supported, modulo null sets, on the finite-orbit part of the action. This extends to stationary actions a basic regularity phenomenon known for measure-preserving actions. We apply this principle to the geometry of Stationary Random Subgroups. For a finitely generated group, we prove that the Schreier graph of a stationary random subgroup has almost surely 0,1,2, or infinitely many ends. Finally, we contrast this probabilistic regularity with the topological notion of Boomerang subgroups: for every k≥3, including k=ℵ0, we construct a Boomerang subgroup of F3 whose Schreier graph has exactly k ends.
In this article, we determine the Hausdorff dimension of the set of exactly ψ-approximable vectors over local fields of positive characteristic. This result is the function field analogue of a recent theorem of Bandi and de Saxcé in the real setting, and extends the main theorem of Zhang to higher dimensions. Our approach adapts the method of Bandi and de Saxcé to the ultrametric setting, enabling us to overcome difficulties arising from the failure of the well-separatedness property for rational functions in higher dimensions.
arXiv:2309.07282v3·Updated Jun 3, 2026·Originally submitted Sep 13, 2023
In this paper, we prove centralizer rigidity near an element of the Weyl chamber flow on a semisimple Lie group. We show that a volume preserving perturbation of an element of the Weyl chamber flow on a quotient G/Γ of an R-split, simple Lie group G either has centralizer of dimension 0 or 1, or is smoothly conjugate to an element of the Weyl chamber flow. We also acquire a general condition for the centralizer of a partially hyperbolic diffeomorphism to be a Lie group.
We propose a novel method for computing invariant densities of certain multidimensional continued fraction algorithms. Inspired by Rauzy induction, our approach builds on the formalism of simplicial systems developed by Fougeron. We introduce a win-lose induction on a graph that is conjugate to the original algorithm, and construct its natural extension by introducing the notion of a dual graph. This method explicitly reconstructs the complete dynamics of the algorithm, yielding a partition of the invariant domain of the natural extension into pieces that map to one another. We further study the ergodic properties of the algorithms within this framework; in particular, we prove that the Modified Triangle algorithm in any dimension admits a unique ergodic measure equivalent to the Lebesgue measure.
The Borel lifting problem asks when a nonsingular near action of a Polish group can be represented by a genuine Borel action. For locally compact Polish groups, a classical theorem of Mackey and Ramsay gives an affirmative answer. At the opposite extreme, Glasner-Tsirelson-Weiss proved that every probability preserving Borel action of a Lévy group is trivial, and asked whether a Lévy group can admit a nontrivial nonsingular Borel action. We prove a fixed-point theorem which gives a negative answer for most of the standard Lévy groups in the literature: if G is a locally Wasserstein group, then every quasi-invariant σ-finite Borel measure for a Borel action of G is supported on the fixed points. The proof introduces Wasserstein stability, a local-to-global principle for compact measure metric groups that refines the Gromov-Milman framework for concentration of measure in topological groups. It asserts for a compact measure metric group that the local total variation of the Haar measure tilted by density kernels, controls global Wasserstein distance from the Haar measure. We show that Wasserstein stability implies concentration, and prove a functional inequality using martingales in the spirit of Milman-Schechtman, which is governed by ℓ1-sums of martingale increments. This gives a geometric criterion by which most of the standard Lévy groups in the literature are Wasserstein or locally Wasserstein, including L0-groups with compact targets, measure preserving and nonsingular automorphism groups, full groups of amenable equivalence relations, isometry groups of Lp-spaces for p=2, the unitary group of the hyperfinite II1-factor, the infinite dimensional unitary and orthogonal groups, the Cameron-Martin affine group, and the isometry group of the Urysohn space.
Alexander Gamburd, Amit Ghosh, Peter Sarnak, Junho Peter Whang
math.NT11J0611E2037A45
We resolve two problems pertaining to indefinite integral ternary quadratic forms, one highlighted by Margulis and the other initiated by Serre, both from 1990. To do so we develop tools for dealing with high ramification in problems involving sums over classes of such forms weighted by their diophantine invariants.
arXiv:2601.09553v3·Updated Jun 1, 2026·Originally submitted Jan 14, 2026
We establish a non-ergodic version of the Host-Kra-Ziegler structure theorem for measure-preserving Zd-actions. Our argument reduces the non-ergodic case to the ergodic theorem (for d≥2 due to Candela and Szegedy) via a measurable selection procedure. We also establish a non-ergodic vertical nilcharacter version of our main result. The non-ergodic version of the Host-Kra-Ziegler structure theorem is a key input in the companion paper by the second author and Fraczyk classifying point processes (i.e. random subsets) of Zd whose law is invariant under the group ASLd(Z) of affine transformations.