arXiv:2508.01912v1

Khintchine-type theorems for weighted uniform inhomogeneous approximations via transference principle

Vasiliy Neckrasov

math.NTmath.DS11J2011J8311J1337A44

Paper metadata

Original submission: August 3, 2025
Claimed MSC: 11J20, 11J83, 11J13, 37A44
arXiv categories: math.NT, math.DS
Paper license: arXiv non-exclusive distribution license ↗
Comments: 28 pages, 1 figure

Abstract

In [Compositio Math. 155 (2019)] Kleinbock and Wadleigh proved a "zero-one law" for uniform inhomogeneous Diophantine approximations. We generalize this statement with arbitrary weight functions and establish a new and simple proof of this statement, based on transference principle. We also give a complete description of the sets of $g$ -Dirichlet pairs with a fixed matrix in this setup from Lebesgue measure point of view. As an application, we consider the set of badly approximable matrices and give a characterization of bad approximability in terms of inhomogeneous approximations. All the aforementioned metrical descriptions work (and sometimes can be strengthened) for weighted Diophantine approximations.

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Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections 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.

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Detailed mathematical audit

Generated August 19, 2026

01Statements3 reported findingsCorrect

The zero–one law for weighted uniform inhomogeneous approximation, the fixed-matrix transference statements, and the weighted badly approximable criteria are correct. The claimed exact continuity reduction in Observation 4.7 is false, but a pair of continuous one-sided envelopes supplies the inclusions actually needed for Theorem 1.3. The explicit formula in Corollary 3.8 is consistent with the corrected reciprocal factor in its proof.

Theorem 1.3 and Proposition 1.7Correct

The weighted zero–one law is correct after replacing exact interpolation by one-sided envelopes

Pages 5, 7, and 24–26 · Theorem 1.3, Proposition 1.7, and proof · arXiv:2508.01912v1

For continuous strictly decreasing error functions, the paper's transference and Fubini argument correctly converts the homogeneous Khintchine-type theorem into the asserted inhomogeneous zero–one law. For an arbitrary non-increasing $g$ , define continuous interpolants on $[l,l+1]$ using the samples $g(l+1),g(l+2)$ for a lower envelope $h_-$ and $g(l-1),g(l)$ for an upper envelope $h_+$ . Then $h_-\leq g\leq h_+$ , and index shifts plus quasimultiplicativity preserve convergence or divergence of the defining series. After the paper's harmless strict-monotonicity perturbation, convergence gives a full-measure subset $\widehat{\mathbf D}[h_-]\subseteq\widehat{\mathbf D}[g]$ , while divergence gives $\widehat{\mathbf D}[g]\subseteq\widehat{\mathbf D}[h_+]$ with the latter null. This verifies Theorem 1.3, and the power-weight substitution gives Proposition 1.7.

Theorems 1.4 and 1.5Correct

The fixed-matrix full and null measure alternatives

Page 6 and pages 18–24 · Theorems 1.4–1.5 and proofs · arXiv:2508.01912v1

Theorem 1.4 follows from the transference-body argument: the absence of arbitrarily small dual points bounds the last minimum of the primal body, whose fixed dilation covers every translate of the lattice. For Theorem 1.5, a sequence of dual approximants produces resonant hyperplanes; outside the null limsup exceptional set, the scalar products with the inhomogeneous shift stay uniformly separated along that sequence, contradicting Dirichlet improvement by the transference inequality. The quasimultiplicative estimates make the constants uniform, so the stated fixed-matrix conclusions and their weighted specializations follow.

Corollary 3.8Correct

The explicit weighted bad-approximability criterion has the stated constants

Pages 16–18 · Corollary 3.8 and proof · arXiv:2508.01912v1

Mahler transference sends a primal bad-approximability constant $b$ to the displayed dual constant and gives the stated full Dirichlet bound $K(b)$ . For the null direction, if the transpose is $(k f_1)$ -approximable, the relation $g(T)=1/f^{-1}(1/T)$ gives $g=k^{-1}f_1$ . Proposition 1.8 therefore yields the factor $\varepsilon^2k^{-1}$ , not the printed $\varepsilon^2k$ . Substituting $k=(bd)^{r_+/(2-r_+)}$ produces exactly the negative powers of $b$ and $d$ displayed in $\kappa(b)$ . Thus the statement is correct and the reciprocal is a uniquely determined proof typo.

02Proofs4 reported findingsContains incorrect or incomplete proofs

Observation 4.7 is false as stated: agreement at integer and denominator-threshold arguments does not preserve the full uniform Dirichlet set across a jump of the error function. A verified envelope argument proves the theorem using inclusions instead of equality. Corollary 3.8 contains a reciprocal typo and one contradictory empty-intersection sign, both fixed uniquely by the surrounding formulas.

Observation 4.7 and proof of Theorem 1.3Incorrect as written · verified repair

Linear interpolation does not preserve Dirichlet sets exactly

Pages 25–26 · Observation 4.7 and final proof of Theorem 1.3 · arXiv:2508.01912v1

Between two consecutive denominator thresholds, the admissible integer denominators are fixed, but a monotone $g$ may jump. A linear interpolant agreeing with $g$ only at the threshold set can demand a smaller error before a new denominator becomes admissible, or allow a larger error after an interior jump. Hence the asserted equality of Dirichlet sets need not hold. The one-sided interpolants $h_-(T)\leq g(T)\leq h_+(T)$ described in the statements audit repair the proof: shifted sums have the same convergence type, and monotonicity gives the two set inclusions needed in the convergence and divergence directions. No exact equality is required.

Proof of Corollary 3.8Typo

The null-measure scaling factor must be reciprocal

Page 18 · last paragraph of the proof of Corollary 3.8 · arXiv:2508.01912v1

For $f(T)=k/T$ , one has $f^{-1}(u)=k/u$ and therefore $g(T)=\frac{1}{f^{-1}(1/T)}=\frac{1}{kT}.$ The application of Proposition 1.8 must consequently read $\widehat{\mathbf D}^{\Theta}[\varepsilon^2k^{-1}f_1]$ , with $\kappa=\varepsilon^2k^{-1}$ . The formula for $\kappa(b)$ in the corollary already uses this reciprocal, so the correction does not change the statement.

Mahler covering stepTypo

A translated-body intersection sign contradicts the next clause

Page 17 · proof of Corollary 3.8 following display (17) · arXiv:2508.01912v1

The application of the covering part of Lemma D says that every translate of the dilated primal body meets the lattice. The display instead ends with an empty intersection and is immediately followed by the statement that it contains a nonzero point when the translate is zero. Replacing $=\varnothing$ by $\neq\varnothing$ is forced by the cited lemma and the following sentence.

Ambient dimensions and weighted exampleTypo

Two notation mismatches have unique corrections

Pages 3 and 8 · distance definition and weighted example · arXiv:2508.01912v1

The nearest-integer definition for a vector in $\mathbb R^m$ minimizes over $\mathbb Z^m$ , not $\mathbb Z^n$ . In the two-coordinate weighted example, the second form contains $q_2$ in the same way the first contains $q_1$ . Both corrections are fixed by dimensions and by the adjacent displayed system and have no effect on any theorem.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2508.01912v1
Authors listed: Vasiliy Neckrasov
Audit date: August 19, 2026

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Khintchine-type theorems for weighted uniform inhomogeneous approximations via transference principle

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The weighted zero–one law is correct after replacing exact interpolation by one-sided envelopes

The fixed-matrix full and null measure alternatives

The explicit weighted bad-approximability criterion has the stated constants

Linear interpolation does not preserve Dirichlet sets exactly

The null-measure scaling factor must be reciprocal

A translated-body intersection sign contradicts the next clause

Two notation mismatches have unique corrections

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