arXiv:2609.36412v1
Twisted Diophantine Approximation II: Uniform Theory
math.NT11J8311J20
Abstract
We develop a general method for uniform twisted metric Diophantine approximation with an arbitrary fixed real matrix. Using all successive minima of the associated diagonal lattice trajectory, we estimate lattice-point counts, accounting for clustering in every direction. Ratios of these counts give exact Hausdorff dimensions of sets of twisted -Dirichlet vectors for a large class of functions , including all except finitely many power functions {, }. We obtain Hausdorff dimension formulae for the endpoint sets obtained by taking the intersection and union of the sets of twisted -Dirichlet vectors over , and prove that taking the intersections causes no dimension drop. Furthermore, we obtain Hausdorff dimensions of level sets of the uniform Diophantine exponent. In addition, we prove zero-full laws for general Hausdorff measures of -Dirichlet sets. Applications recover the one-dimensional Kim-Liao formulae, strengthen Aggarwal's escape-of-mass upper bound, reproduce the metric criteria of Moshchevitin and the second author and give certain extensions of the above statements.
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Proof lineage
Twisted Diophantine Approximation II: Uniform Theory
A statement-restricted recursive graph for the proof-critical inputs behind the pressure formula, the no-dimension-drop theorem, the Hausdorff-measure zero-full law, the sharp escape-of-mass construction, and the transference applications. Contextual comparisons, historical attributions, and results merely recovered by the focal paper are excluded. Books are terminal sources.
Open dependence graph →AI-generated audit
Audit summary
Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.
Current report
Detailed mathematical audit
01Statements7 reported findingsNo statement issues found
The pressure formulae for the lower and upper endpoint sets, the no-dimension-drop identity, the near-power and uniform-exponent formulae, the Hausdorff-measure zero-full law, the one-dimensional continued-fraction formulae, the escape-of-mass estimates and sharpness statement, and the transference applications are correctly stated in arXiv:2609.36412v1. The exceptional power set and every required regularity or nonemptiness qualification are carried into the conclusions. No mathematical statement requiring correction was found.
Theorem 1.2No issue found
The pressure formula and no-dimension-drop identities have the stated scope
PDF pages 5–6 and 31–39 · Theorem 1.2, Theorem 5.1, Proposition 5.3, and proof · arXiv:2609.36412v1
For a continuous strictly decreasing lower-quasimultiplicative function, the equal-radius mesh gives uniform upper and lower estimates for every constant multiple. Taking the infimum and supremum in the constant and then letting the mesh parameter tend to infinity produces the two endpoint pressures. The variable-scale Cantor construction supplies a single function whose dimension approaches the lower pressure, so the intersection defining loses no dimension. The union identity is countable after restricting constants to integers. These facts also prove that constant-rescaling invariance is equivalent to equality of the two pressures.
Theorem 1.4; Proposition 1.6; Corollary 1.7No issue found
Near-power and uniform-exponent dimensions include the critical qualifications
PDF pages 7–8 and 39–48 · Section 6 · arXiv:2609.36412v1
Away from the finite set , threshold crossings of the normalized successive minima have zero density. This makes the pressure independent of the constant and converts it into the stated rank integral. Comparison with nearby powers proves the same formula for functions of limiting power order. At every exponent, including critical ones, monotonicity of the approximation sets and the no-drop construction give the displayed superlevel, strict-superlevel, and conditional exact-level formulae.
Theorem 1.9No issue found
The zero-full alternatives are correctly formulated on the orbit-closure group
PDF pages 9–10 and 48–51 · Theorem 1.9 and proof · arXiv:2609.36412v1
Every approximable shift lies in , and the proof works with its intrinsic Haar and Hausdorff measures. Translation by the dense subgroup amplifies any positive finite Hausdorff measure to a Haar-conull invariant set; infinite measure gives local fullness directly. Difference and sum constructions then establish exact amplification and show that the two endpoint sets are Borel subgroups. The zero-dimensional finite-group case is separated explicitly.
Propositions 1.10–1.11No issue found
The reversed endpoint representations and dimension continuity are valid
PDF pages 10–11 and 51–53 · Propositions 1.10–1.11 and proofs · arXiv:2609.36412v1
Sparse piecewise-linear interpolation constructs a single smaller function for each point in the singular endpoint and a single larger obstruction for each point outside the upper endpoint. The diagonal pressure construction can be carried out within the lower-quasimultiplicative class, which gives the stated supremum and infimum identities for the endpoint dimensions rather than only the set-theoretic union and intersection formulas.
Theorem 2.2No issue found
The one-dimensional formulae correctly specialize the general pressure
PDF pages 13–15 and 53–62 · Theorem 2.2 and proof · arXiv:2609.36412v1
The continued-fraction profile identifies the two active branches of the lattice count and isolates the convergent indices at which they cross the approximation curve. Logarithmic sparsity makes discarded transition intervals negligible. Telescoping the remaining pressure increments produces the two printed products in the slow and fast regimes, including the finite-index cases of dimensions one and zero. Multiplying or argument-rescaling shifts the profile by only a bounded amount and leaves the dimension unchanged.
Propositions 2.4–2.5 and 2.7No issue found
The escape-of-mass upper bounds and sharpness conclusion are correct
PDF pages 16–17 and 62–69 · Propositions 2.4–2.5, 2.7, and proofs · arXiv:2609.36412v1
The rank integral is bounded by the time spent below a first-minimum cutoff and by the product of all active minima. At the critical exponent, Minkowski's product control leaves at most active factors and yields the improved coefficient . For sharpness, the tent template has active proportions and , while the zero phase has proportion ; the template-realization theorem produces a matrix with exactly that escape proportion. The pressure calculation and monotonicity in the approximation constant give the matching lower bound, so every occurs.
Propositions 2.8–2.10No issue found
The transference and trivial-shift conclusions preserve all quantifiers
PDF pages 17–18 and 69–71 · Propositions 2.8–2.10 and proofs · arXiv:2609.36412v1
A sequence of good dual approximants confines the uniformly approximable shifts to character strips of measure , and the zero-full theorem turns positive measure at a smaller constant into the contradiction needed for nullity. Conversely, the finite-scale transference lemma with produces an admissible denominator for every shift when the transpose is not approximable. Adding a fixed positive vector gives the positive-denominator variant with the factor three. The telescoping difference argument for two denominators proves that sufficiently small approximation sets contain exactly the prescribed orbit shifts.
02Proofs8 reported findingsNo proof issues found
All central proof chains close in the exact source. The audit followed the lattice-count comparison into both covering constructions, the pressure and variable-scale diagonal arguments, the normalized-minima analysis for powers, the compact-group zero-full argument, the continued-fraction specialization, the escape-of-mass template construction, and both transference directions. Constants, directions of inclusions, limiting operations, and exceptional cases agree with the stated conclusions. No incorrect or materially incomplete proof was found.
Lemmas 4.1–4.4No issue found in the reviewed proof
The successive-minima profile gives the required lattice counts
PDF pages 22–24 · Section 4.1 · arXiv:2609.36412v1
After conjugating the box to a cube, the companion-paper count is exactly the product defining up to a dimension-only constant. Fixed dilations preserve the count up to such a constant, and translated boxes contain no more lattice points than the corresponding centered difference box. The monotonicity and scaling estimates used later follow with the displayed exponents.
Propositions 3.1–3.2No issue found in the reviewed proof
The upper and lower covering constructions have matching transition counts
PDF pages 20–31 · Sections 3–4 · arXiv:2609.36412v1
The upper tree uses maximal disjoint families whose triples cover every omitted ball; the containment margins place all relevant lifted centers in the numerator box, while separation supplies the denominator count. The lower tree reserves both approximation and denominator margins, giving the reverse transition estimate. The condition bridges every parameter between mesh points, and the mass distribution bound controls arbitrary intermediate radii rather than only construction scales.
Theorem 5.1 and Proposition 5.3No issue found in the reviewed proof
Uniform pressure estimates and the no-drop diagonalization close
PDF pages 31–39 · Section 5 · arXiv:2609.36412v1
Lower quasimultiplicativity makes consecutive denominator bounds and radii satisfy the construction hypotheses uniformly once is large. The fixed branching constants contribute only , uniformly in the rescaling constant. For no drop, the block lengths dominate every preceding transition while their excess lengths are sublinear; the interpolated function remains decreasing and lower quasimultiplicative, is , and retains the pressure lower bound after the transition blocks are removed.
Section 6No issue found in the reviewed proof
The near-power pressure and exponent-level arguments handle all threshold cases
PDF pages 39–48 · Lemmas 6.1–6.6 and Proposition 1.6 · arXiv:2609.36412v1
Exterior-power covolumes control sums of logarithmic minima. When an exponent is outside , a separated-block argument shows that every zero threshold has density zero. On the remaining mesh rectangles, the active exterior subgroup and vertical slope are constant, so the pressure telescopes to the rank identity with an error uniform in the constant. Order comparison with neighboring powers gives both one-sided exponent limits; the variable-power no-drop lemma supplies the exact-level lower bound whenever that level is nonempty.
Theorem 1.9 and Propositions 1.10–1.11No issue found in the reviewed proof
The invariant-measure and endpoint interpolation arguments are complete
PDF pages 48–53 · Section 7 · arXiv:2609.36412v1
The proof treats separately zero, finite positive, and infinite Hausdorff measure. In the finite case, dense-subgroup invariance extends to and uniqueness of Haar measure makes the invariant set conull; comparison of the dimension function with the intrinsic power measure supplies local fullness. Translation and subtraction of witnesses give the factors used in exact amplification and subgroup closure. The sparse interpolation sequences used for the endpoint identities preserve monotonicity and the required little- relations.
Theorem 2.2No issue found in the reviewed proof
The continued-fraction decomposition and telescoping products agree
PDF pages 53–62 · Section 8 · arXiv:2609.36412v1
Successive convergents determine the first lattice minimum on each branch interval, and unimodularity supplies the second. The relevant crossing times are ordered by the slow- or fast-decay hypothesis. Boundary blocks have zero logarithmic density, while the retained intervals contribute exactly the numerator factors appearing in formulas (2.7) and (2.10). The pressure normalization matches the final denominator .
Propositions 2.4–2.5 and 2.7No issue found in the reviewed proof
The rank bounds and sharp template calculation give the advertised coefficient
PDF pages 62–69 · Section 9 · arXiv:2609.36412v1
The profile product identity separates the rank term from the normalized lattice-count term. Shrinking the cutoff and integrating yields the general upper bound; at the critical exponent the first-minimum escape set controls every active factor and produces . In the sharp template, cycle count and unfinished-cycle length are sublinear, so both escape frequency and pressure are determined by the three printed phase proportions. For constants outside the initially normalized range, the zero-phase contribution is nonnegative and monotonicity of the approximation sets gives the same endpoint lower bound.
Propositions 2.8–2.10No issue found in the reviewed proof
The dual-strip, full-space, and telescoping arguments are valid
PDF pages 69–71 · Section 10 · arXiv:2609.36412v1
The scalar-product identity converts simultaneous primal and dual inequalities into a strip condition with the correct factor. Haar pushforward gives its exact measure, and Fatou's inequality in the required direction bounds the liminf. The imported finite-scale transference lemma is applied with the dual inverse functions at matching parameters. For trivial shifts, two eventual witnesses subtract to a homogeneous approximation that the hypothesis forbids unless the denominator difference is zero, forcing a single orbit point.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.
04Sources6 reported findingsNo source-use issues found
Every direct proof-critical citation was located and checked in an exact public version or a disclosed proof-bearing counterpart. The lattice-count input from the companion paper agrees with Henk's two underlying results; the mass-distribution, compact-group, template-realization, Haar-pushforward, and inhomogeneous-transference inputs all have the hypotheses and conclusions required by the focal proof. Recursive non-book dependencies are represented in the graph, and the three published books are explicit terminal sources.
Companion Lemma 5.1 and Henk's lattice-point boundsApplicable and sufficient
The counting input matches the focal profile exactly
Focal PDF pages 22–23 · companion arXiv:2609.36411v1 pages 14–15 · Henk Lemma 2.1 and Theorem 1.5 · arXiv:2609.36412v1
The companion lemma compares the centered box count with after the focal diagonal normalization. Henk Lemma 2.1 yields the fixed-dilation comparison through lattice cosets, and Theorem 1.5 yields the required upper product. The companion paper proves the matching lower bound internally by taking bounded integer combinations of linearly independent minimizing vectors.
Kim and Neckrasov, arXiv:2609.36411v1 ↗Beresnevich--Velani Mass Distribution PrincipleApplicable and sufficient
The cited principle directly proves the lower-dimension step
Focal PDF pages 27–30 · source Annals page 975 · arXiv:2609.36412v1
The source states the principle for a probability measure supported on a metric-space subset and proves it by applying the ball estimate to an arbitrary fine cover. Taking gives exactly the focal Lemma 4.8, with no Euclidean-only hypothesis missing.
Beresnevich and Velani, Annals of Mathematics 164 (2006), 971–992 ↗Kleinbock--Shah--Starkov and LeeApplicable and sufficient
The compact-group inputs support the zero-full proof
Focal PDF pages 9 and 48–49 · KSS Lemma 1.1.3 · Lee Theorem 20.12 · arXiv:2609.36412v1
Lee's Closed Subgroup Theorem makes a compact Lie subgroup of the torus. KSS Lemma 1.1.3 says that a finite measure's stabilizer is closed; continuity of translation on measures proves the assertion immediately. Since is dense in , these facts justify the extension from discrete translation invariance to Haar invariance used in Theorem 1.9.
Kleinbock, Shah, and Starkov, Handbook of Dynamical Systems 1A (2002) ↗Das--Fishman--Simmons--Urbański template realizationApplicable and sufficient
The realization theorem covers the sharpness template
Focal PDF pages 65–67 · source Definition 4.1 and Theorem 4.2 · arXiv:2609.36412v1
The focal tent functions are ordered, balanced, have allowed coordinate slopes, and satisfy the convexity and quantized-slope rule at their only strict gap. The source theorem realizes every such template within bounded error as the successive-minima function of a matrix; its proof includes the zero-average-contraction case relevant here. This is precisely the existence statement needed for Proposition 2.5. The checked source is the final public preprint arXiv:1901.06602v13 corresponding to the published article; a separate publisher PDF was not checked.
Das, Fishman, Simmons, and Urbański, final public preprint arXiv:1901.06602v13 ↗Halmos §63, Theorem CApplicable and sufficient
Haar pushforward gives the exact character-strip measure
Focal PDF page 69 · Lemma 10.1 · arXiv:2609.36412v1
For every nonzero integer vector , the character , , is a continuous surjective homomorphism. Halmos' theorem identifies the pushforward of normalized Haar measure with normalized Haar measure on the circle, so the inverse image of the interval has measure when .
Paul R. Halmos, Measure Theory, §63 ↗Moshchevitin--Neckrasov Lemma IApplicable and sufficient
The transference constant and normalization agree
Focal PDF pages 70–71 · source arXiv:2503.21180v3 page 28 · arXiv:2609.36412v1
Lemma I assumes a uniform lower bound for nonzero transpose vectors of norm at most and produces, for every shift, a primal denominator of norm at most with error at most . The focal variables are substituted in the same order and use the same constant . The source explicitly derives this lemma from Bugeaud--Laurent Lemma 3, whose proof invokes Cassels Chapter V, Theorem XVII(B); that recursive chain is represented in the graph.
Moshchevitin and Neckrasov, arXiv:2503.21180v3 ↗