MathAudit

Paper metadata

Original submission
January 20, 2012
Latest revision
February 22, 2012 · version 2
Claimed MSC
11J13
arXiv categories
math.NT
Comments
Short communication

Abstract

We give some comments on W.M. Schmidt's theorem on Diophantine approximations with positive integers and our recent results on the topic.

AI-generated audit

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 20, 2026
01Statements2 reported findingsCorrect

The optimized positive-integer exponent bound and the comparison with the counterexample are correct.

Theorem 2Correct

Optimized lower bound for the positive exponent

Pages 2-3 · equations (5)-(6) and Theorem 2 · arXiv:1201.4232v2

Optimizing Schmidt's two alternatives under yω^−1≤z≤yω/ω^y^{\widehat\omega-1}\leq z\leq y^{\omega/\widehat\omega} produces the two regimes A1,A2A_1,A_2. Their boundary agrees, and the resulting maximum is exactly the displayed pair G(ω)G(\omega) and ω^−1+ω^/ω\widehat\omega-1+\widehat\omega/\omega.

Full paper, version 2 ↗
Section 2Correct

The counterexample exponent comparison is consistent

Page 3 · Theorem 3 and the calculation following it · arXiv:1201.4232v2

The stated values of ω\omega and ω^\widehat\omega put the constructed pair in A2A_2, and direct substitution gives the weaker universal bound quoted there. The lower estimate in Theorem 3 gives ω+≤σ\omega_+\leq\sigma, while the approximating subsequence in the cited construction supplies the reverse inequality, hence ω+=σ\omega_+=\sigma.

02Proofs2 reported findingsCorrect

The note's optimization and exponent substitutions are correct; its external inputs are explicitly identified as previously proved results.

Equation (5)Correct and complete

The minimax reduction has the correct admissible ranges

Page 2 · equation (5) · arXiv:1201.4232v2

Jarnik's inequality gives the lower endpoint for the relation between ordinary and uniform exponents. For fixed y,zy,z, the two monomials in xx move in opposite directions, so the minimum is optimized at their intersection or an endpoint. Those alternatives give exactly the two formulas stated after the partition of AA.

Equations (1)-(2)Correct and complete

The classical implications are used in the correct direction

Pages 1-2 · equations (1)-(2) · arXiv:1201.4232v2

Taking the minimum of the two terms in Schmidt's bound yields ω+≥φ\omega_+\geq\varphi at the fixed point. The same argument with uniform rather than liminf control gives ω^+≥ω/(ω−1)\widehat\omega_+\geq\omega/(\omega-1); no interchange of ordinary and uniform quantifiers is made.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1201.4232v2
Authors listed
Nikolay G. Moshchevitin
Audit date
August 20, 2026
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