MathAudit

Paper metadata

Original submission
June 1, 2023
Latest revision
January 27, 2024 · version 2
Claimed MSC
11J20, 11J83, 28A78
arXiv categories
math.NT
Paper license
CC BY 4.0 ↗
Comments
11 pages; Theorem 1.7 and its proof are revised

Abstract

Kurzweil's theorem ('55) is concerned with zero-one laws for well approximable targets in inhomogeneous Diophantine approximation under the badly approximable assumption. In this article, we prove the divergent part of a Kurzweil type theorem via a suitable construction of ubiquitous systems when the badly approximable assumption is relaxed. Moreover, we also discuss some counterparts of Kurzweil's theorem.

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Audit summary

Audited against arXiv v2

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.

Current report

Detailed mathematical audit

Generated August 19, 2026
01Statements3 reported findingsCorrect

The Hausdorff-measure Kurzweil theorem, its badly approximable consequences, and the sufficient condition for full measure are correct, with one harmless range correction in an auxiliary proposition.

Theorem 1.3Correct

The Hausdorff zero-full criterion is correct

Pages 2 and 5–9 · Theorem 1.3 · arXiv:2306.00847v2

The convergence half follows from Hausdorff–Cantelli on the return balls. For divergence, the best-approximation scales define a locally ubiquitous system with the announced radius; the ubiquity theorem converts divergence of the same series into full Hausdorff measure.

Full paper, version 2 ↗
Proposition 5.1Minor formal correction

The parameter must satisfy α>n\alpha>n

Page 9 · Proposition 5.1 · arXiv:2306.00847v2

The proposition is printed for every α>0\alpha>0 but concludes inclusion in Bad⁡A((α−n)/m)\operatorname{Bad}_A((\alpha-n)/m), while Bad⁡A(δ)\operatorname{Bad}_A(\delta) was defined only for δ>0\delta>0. Its proof yields the positive lower bound (α−n)/m(\alpha-n)/m precisely when α>n\alpha>n, and Theorem 1.7 applies it only with α=mδ+n>n\alpha=m\delta+n>n. Replace the proposition’s range by α>n\alpha>n; no downstream statement changes.

Theorem 1.7Correct

The full-measure criterion follows from the best-approximation sequence

Pages 4 and 9–10 · Theorem 1.7 · arXiv:2306.00847v2

The summability hypothesis makes the exceptional events ∥b⋅yk∥Z≤αγk\|\mathbf b\cdot\mathbf y_k\|_{\mathbb Z}\leq\alpha\gamma_k summable. Borel–Cantelli places almost every target in the eventual lower-bound set, and Proposition 5.1 with α=mδ+n\alpha=m\delta+n gives membership in Bad⁡A(δ)\operatorname{Bad}_A(\delta).

02Proofs2 reported findingsCorrect

The Hausdorff–Cantelli, ubiquity, transference, and Borel–Cantelli arguments are correct and complete in their operative ranges.

Sections 3–4Correct and complete

The convergence and divergence proofs use the same scale sequence

Pages 5–9 · Sections 3–4 · arXiv:2306.00847v2

The selected return times have a uniform contraction ratio, the resonant balls cover a fixed proportion at each scale, and the series in the ubiquity theorem is exactly the one in Theorem 1.3 up to fixed constants.

Section 5Correct and complete

The sufficient-condition proof closes

Pages 9–10 · Section 5 · arXiv:2306.00847v2

The intervals [Uk,Vk)[U_k,V_k) cover every sufficiently large denominator scale, and the transference inequality gives the required lower bound after the range correction α>n\alpha>n. The summability assumption then supplies the almost-everywhere conclusion.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:2306.00847v2
Authors listed
Taehyeong Kim
Audit date
August 19, 2026
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