arXiv:2609.07506v1

Solutions to 6 problems in recurrence and Van der Corput sets

math.DSmath.NT37A4405D1011K0637A46

Paper metadata

Original submission: September 7, 2026
Claimed MSC: 37A44, 05D10, 11K06, 37A46
arXiv categories: math.DS, math.NT
Comments: 20 pages

Abstract

We present solutions to six problems concerning sets of nice recurrence and van der Corput (vdC) sets. We answer a question of Moreira, repeated by Fish and Skinner, by showing that every set of ergodic nice recurrence is a set of nice recurrence. We answer questions of Bergelson as well as Bergelson and Lesigne by showing that every set of nice recurrence is a nice vdC-set, that the family of nice vdC-sets is partition regular, and that the family of sets of nice recurrence is partition regular. We mention that Chapman independently proved that sets of nice recurrence are partition regular. We answer a question of Kelly and Lê, a special case of which was previously asked by Peres, and show that for every nontrivial compact metrizable group $K$ , every $K$ -vdC set is also a vdC set. Lastly, we answer a question of Farhangi, Rodríguez, and Tucker-Drob and show that for any countably infinite discrete group $G$ , a vdC set in $G$ is a set of operatorial recurrence in $G$ . The initial solutions to these problems were produced by an AI system. The authors checked their correctness, simplified and reformulated them as clear, human-readable proofs, and situated the resulting contributions within the existing literature.

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Generated October 24, 2026

01Statements4 reported findingsNo statement issues found

The equivalence of the recurrence formulations, partition regularity, compact-group van der Corput equivalence and passage from bounded to square-integrable operatorial recurrence are supported under their stated countability, amenability and metrizability hypotheses. The entire manuscript, including the auxiliary constructions and remarks used by the proofs, was reviewed.

Theorem 2.2No issue found

Ergodic nice recurrence and the positive-definite criterion are equivalent

PDF pp. 4–6 · Proposition 2.1 and Theorem 2.2 · arXiv:2609.07506v1

For a nonempty subset $R$ of a countably infinite group, failure of the real positive-definite criterion gives a normalized function $\varphi$ uniformly negative on $R$ . Its ergodic real $L^2$ realization can be approximated by a bounded real function with correlation error at most $\eta(2+\eta)$ , uniformly over every group element, because the Koopman operators are isometries. Subtracting the mean only decreases these real correlations. Rescaling to $h=\frac12+\delta F\in[0,1]$ gives uniformly suboptimal two-point correlations. The independent-coordinate threshold construction in Proposition 2.1 realizes these as intersections of sets; the relevant coordinates are distinct since the identity cannot belong to a set on which $\varphi$ is negative. Its Bernoulli factor is weakly mixing for an infinite group, so the realization remains ergodic. Conversely, splitting a representation vector as $f=Pf+f_0$ makes $g\mapsto\operatorname{Re}\langle\pi(g)f_0,f_0\rangle$ positive definite, and the criterion yields the required lower bound on the modulus of its full correlation. Applying this to Koopman representations and then to indicators closes the equivalence chain. In the abelian case, the trivial-character mass of the spectral measure is $\|Pf\|^2$ , and the bounded spectral realization used in the other direction has zero mean after removing that atom. Adding the constant component of size $\sqrt a$ gives exactly the asserted nice Fourier criterion. The absolute values in the operatorial and nice van der Corput formulations were checked against the rendered PDF; no unjustified lower bound on a complex number is used.

Theorem 3.3 and Corollary 3.4No issue found

The positive-coefficient polynomial gives partition regularity

PDF pp. 7–8 · Lemmas 3.1–3.2, Theorem 3.3 and Corollary 3.4 · arXiv:2609.07506v1

The sum and product of positive-definite functions are positive definite by direct sums and tensor products, so every polynomial with nonnegative coefficients preserves this property. If two bounded real witnesses have bounds in $[-M,M]$ and one value is below $-\varepsilon$ , then $x(2M+y)^K+y(2M+x)^K$ is uniformly negative for sufficiently large fixed $K$ . When $y\le0$ , the first term is at most $-\varepsilon M^K$ and the second is nonpositive. When $y\ge0$ , the positive second term is at most $M(2M-\varepsilon)^K$ , which is at most half the magnitude $\varepsilon(2M)^K$ of the negative first term by the chosen $K$ . Thus one gets a single positive-definite witness uniformly negative on the union of the two exceptional sets. The criterion in Theorem 2.2 applies directly, and finite partition regularity follows by induction. This does not rely on multiplying two negative witnesses, which would give the wrong sign.

Theorems 4.6 and 4.8No issue found

The compact-group conclusions hold for noncommutative compact groups as well

PDF pp. 8–15 · Lemmas 4.2–4.3, Proposition 4.5 and Theorems 4.6–4.8 · arXiv:2609.07506v1

Lemma 4.2 turns a finite-dimensional vector witness into a scalar one using a finite random vector $\xi$ satisfying $\mathbb E\xi=e_1$ and $\mathbb E\xi\xi^*=I$ . Both identities hold for its displayed $2(d-1)$ equally weighted values. The real version and an independent-coordinate threshold extension give a sign-valued witness with zero correlations and nonzero mean whenever the ordinary van der Corput property fails. For the forward compact-group implication, apply the vector criterion to $U(F^{-1})\xi$ for each nontrivial irreducible unitary representation $U$ ; the ratio in its scalar product is the inverse of the assumed Haar-distributed ratio. Peter–Weyl then identifies the law of $F$ with Haar measure. For the converse, take conditional coordinate densities $1+s(T_gx)\psi$ with real nonzero Haar-mean-zero $\psi$ . Invariance of the resulting kernel product measure follows from the displayed right-coordinate permutation. For each $h\ne e$ , conditional independence makes the ratio density $1+s(T_hx)s(x)(\psi*\check\psi)$ ; averaging removes its second term exactly. The coordinate marginal is instead $1+m\psi$ , with $m\ne0$ , so it is not Haar. The convolution order is valid without commutativity. The uniform-distribution formulation additionally assumes amenability and a fixed left-Følner sequence. Its inverse-correspondence construction and its shift-empirical-measure argument preserve these precise left/right conventions, so neither metrizability nor the prescribed averaging sequence is silently changed.

Theorem 5.3No issue found

A Gaussian sign witness proves the bounded-to-square-integrable implication

PDF pp. 15–18 · Lemmas 5.1–5.2, Theorem 5.3 and Remark 5.5 · arXiv:2609.07506v1

An $L^2$ witness with nonzero mean $a$ gives a positive-definite centered correlation equal to $-|a|^2$ on $H$ . Conversely, for a real positive-definite $\varphi$ equal to a fixed $r<0$ on $H$ , write $A=\varphi(e)>0$ . The invariant Gaussian coordinate law with covariance $\varphi(g^{-1}k)$ is consistent even when a finite covariance matrix is singular. The coordinate signs have zero mean and common correlation $\lambda=(2/\pi)\arcsin(r/A)<0$ . Adjoining a fixed point with weights $1/(1-\lambda)$ and $-\lambda/(1-\lambda)$ gives a bounded sign function whose correlations vanish exactly on $H$ and whose mean is $-\lambda/(1-\lambda)>0$ . The endpoint $r=-A$ is allowed: the Gaussian pair is then perfectly anticorrelated and the same formulas remain valid. The optional shifted-sign construction also works by continuity of the common bivariate Gaussian law. To relate the formulation to arbitrary unitary representations, a vector with invariant projection $P\xi\ne0$ and correlations zero on $H$ gives the positive-definite function of $\xi-P\xi$ with constant value $-\|P\xi\|^2$ there, and hence the same bounded witness. Thus the announced operatorial conclusion is obtained without requiring every positive-definite function to have a bounded realization.

02Proofs2 reported findingsNo proof issues found

The correlation-preserving threshold, scalarization, kernel product and Gaussian constructions were verified. The material external realization and inverse-correspondence inputs have exactly the hypotheses required here.

Theorem 2.2: realization inputsNo issue found in the reviewed proof

The ergodic and abelian spectral realizations apply in the required categories

PDF pp. 5–6 · cited FTD26 Theorem 2.8 and Lemma 3.4 · arXiv:2609.07506v1

The primary source Farhangi–Tucker-Drob, Asymptotic dynamics on amenable groups and van der Corput sets, Theorem 2.8, supplies an ergodic $L^2$ realization for positive-definite functions on locally compact second countable groups, with real functions for real input. Countable discrete groups satisfy this scope; amenability is not a hypothesis of that result. The underlying construction splits the cyclic representation into compact and weakly mixing parts, realizes the former on its compact closure and the latter by a weakly mixing Gaussian system, and takes their product. For real input the same construction may be made in the real GNS Hilbert space. Lemma 3.4 supplies the bounded unit-modulus abelian realization and its mean, so the atom-removal calculation on p. 6 is valid. The compact-closure realization and the independent Gaussian component also explain why merely taking an arbitrary Gaussian system would not suffice to obtain ergodicity. All uniform approximation estimates and all mean/correlation calculations needed after these inputs are given in the manuscript and were checked directly.

Theorem 4.8 and Section 5No issue found in the reviewed proof

The inverse correspondence and probability constructions close the remaining proof chains

PDF pp. 13–18 · Theorem 4.8 and Lemmas 5.1–5.2 · arXiv:2609.07506v1

The primary Rodríguez Martín inverse-correspondence theorem, Theorem 2.7, concerns an arbitrary countably infinite amenable group, its specified Følner sequence, a compact subset $D\subset\mathbb C$ and a measurable $D$ -valued observable, with all continuous finite-coordinate test functions. Taking $D$ to be the Cantor set therefore covers the Borel lift used here, even for nonergodic systems; no pointwise ergodic theorem along a non-tempered sequence is being presumed. Composing its test functions with the continuous Cantor surjection yields exactly both Haar-ratio and marginal averages. Conversely, compact metrizability makes the shift probability space sequentially compact, and left-Følner invariance makes each empirical cluster measure invariant. Continuity of the coordinate and ratio maps then forces every marginal cluster law to be Haar, proving convergence of the full marginal sequence. In Section 5, positive definiteness makes the finite Gaussian covariance matrices positive semidefinite; principal-submatrix consistency and countability give the asserted unique product-space probability measure and shift invariance. The arcsine sign identity can also be checked independently by representing a standardized pair as two projections of a rotationally invariant planar Gaussian: the angular sector where their signs disagree has fraction $\arccos(r/A)/\pi$ . This verifies the needed probability formula, including its degenerate endpoints, without an additional hypothesis on Gaussian densities.

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Paper: arXiv:2609.07506v1
Authors listed: Sohail Farhangi, Saúl Rodríguez Martín
Audit date: October 24, 2026

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Solutions to 6 problems in recurrence and Van der Corput sets

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Ergodic nice recurrence and the positive-definite criterion are equivalent

The positive-coefficient polynomial gives partition regularity

The compact-group conclusions hold for noncommutative compact groups as well

A Gaussian sign witness proves the bounded-to-square-integrable implication

The ergodic and abelian spectral realizations apply in the required categories

The inverse correspondence and probability constructions close the remaining proof chains

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