MathAudit

Paper metadata

Original submission
January 19, 2015
Latest revision
March 17, 2015 · version 2
Claimed MSC
arXiv categories
math.NT
Comments
8 pages. Updated 17/03/15 to incorperate some very helpful observations from Sanju Velani

Abstract

We prove a result in the area of twisted Diophantine approximation related to the theory of Schmidt games. In particular, under certain restrictions we give a affirmative answer to the analogue in this setting of a famous conjecture of Schmidt from Diophantine approximation.

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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 20, 2026
01Statements2 reported findingsCorrect

The weighted twisted badly approximable set is correctly shown to be one-half winning under the stated homogeneous badness hypothesis. The countable-intersection corollary has an evident notation mismatch in its hypothesis.

Theorem 1.1Correct

The winning conclusion is verified

Pages 3-8 · Theorem 1.1 and its proof · arXiv:1501.04433v2

Transference turns the assumption L∈Bad(k,n,m)L\in\mathrm{Bad}(k,n,m) into a uniform dual lower bound. The chosen dual best approximations split into finitely many Euclidean-lacunary subsequences; the cited lacunary-sequence game lemma makes their common avoidance set one-half winning, and the final scalar-product estimate embeds that set in BadL(k,n,m)\mathrm{Bad}_L(k,n,m).

Corollary 1.2Typo · uniquely repairable

The hypothesis uses a single undefined weight instead of the displayed sequence

Page 3 · Corollary 1.2 · arXiv:1501.04433v2

For a sequence ktk_t, the matrix must belong to ⋂tBad(kt,n,m)\bigcap_t\mathrm{Bad}(k_t,n,m) before Theorem 1.1 can be applied for every tt. The printed hypothesis instead says L∈Bad(k,n,m)L\in\mathrm{Bad}(k,n,m) with an undefined single kk. The intended correction is forced by the conclusion and by countable stability of winning sets.

02Proofs3 reported findingsCorrect

The transference, lacunarity, and Schmidt-game steps form a complete proof after harmless local notation corrections.

Lemma 2.2Correct

The dual best-approximation sequence has a finite lacunary partition

Pages 5-6 · Lemma 2.2 · arXiv:1501.04433v2

The homogeneous badness inequality bounds the first weighted coordinate below in terms of the scale parameter. Comparing parameters a fixed number of steps apart therefore multiplies the Euclidean norm by at least two, so residue classes of indices give finitely many two-lacunary subsequences.

Lemma 2.3 and Corollary 2.4Correct

Lacunary avoidance transfers to the target set

Pages 6-8 · Lemma 2.3, Corollary 2.4, and final proof · arXiv:1501.04433v2

Each lacunary subsequence yields a one-half winning set, and finite intersection preserves that parameter. For an avoided target, the identity pairing a primal approximation with the selected dual vector gives a positive uniform lower bound on the weighted inhomogeneous error for every nonzero integer vector.

Local references and scalar inequalityTypos

Two labels and one coefficient domain need mechanical correction

Pages 6-7 · text following Lemma 2.3 and proof of Theorem 1.1 · arXiv:1501.04433v2

The reference to Lemma 2.4 must be Lemma 2.3. The inequality ∥az∥≤∣a∣∥z∥\|az\|\leq|a|\|z\| is asserted for real aa but is true and used here for integer aa; replacing a∈Ra\in\mathbb R by a∈Za\in\mathbb Z is the unique relevant correction.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1501.04433v2
Authors listed
Stephen Harrap, Nikolay Moshchevitin
Audit date
August 20, 2026
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