MathAudit

Paper metadata

Original submission
May 20, 2023
Claimed MSC
11J83, 11J20, 11J13
arXiv categories
math.NT
Comments
4 pages, short communication

Abstract

We prove an easy statement about inhomogeneous approximation in metric theory of Diophantine Approximation.

AI-generated audit

Audit summary

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.

Current report

Detailed mathematical audit

Generated August 20, 2026
01Statements3 reported findingsCorrect

For every well-distributed sequence, the metric inhomogeneous approximation conclusion of Theorem 1 is correct.

Well-distributed scalesCorrect and complete

A positive proportion of separated boxes receives sequence points

Pages 2–3 · box selection preceding equations (6)–(8) · arXiv:2305.12230v1

At every selected well-distributed scale, all interior grid boxes contain a sequence point. Restricting to coordinate indices congruent to one modulo three makes the smaller target boxes pairwise disjoint while retaining q′≍Cqq'\asymp_C q centers. This yields the uniform positive lower measure used in the limsup argument.

Theorem 1Correct

Almost every target receives the claimed infinitely many approximations

Pages 2–4 · Theorem 1 and its proof · arXiv:2305.12230v1

At each selected scale, well distribution supplies one sequence point in a positive proportion of separated boxes. The total measures diverge, while the inductive choice of scales gives the required quasi-independence bound. The second Borel–Cantelli lemma then yields the liminf conclusion for almost every target.

Full paper, version 1 ↗
Box enumerationTypo · no status impact

One upper index contains an extra power

Page 2 · grid-box enumeration preceding equation (6) · arXiv:2305.12230v1

The coordinate indices run up to the one-dimensional grid count in each direction; the printed upper bound carries an extra nnth power. The next line's total number of boxes and every later count use the correct coordinate bound.

02Proofs1 reported findingCorrect

The separated-box construction and quasi-independent Borel–Cantelli argument are correct and complete after the grid-index typo is repaired.

Proof of Theorem 1Correct and complete

The metric limsup argument closes

Pages 2–4 · equations (6)–(13) · arXiv:2305.12230v1

The selected boxes at a fixed scale are disjoint, their union has measure comparable to the prescribed divergent term, and later scales are chosen to control every earlier intersection. Repeated sequence points form at most a null exceptional set and do not affect the conclusion.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:2305.12230v1
Authors listed
Nikolay Moshchevitin
Audit date
August 20, 2026
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  2. 02Submit evidenceIdentify the finding and give a formal mathematical response.
  3. 03Re-evaluateA separate agent checks the response and records a disposition.
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