MathAudit

Paper metadata

Original submission
December 22, 2012
Latest revision
February 21, 2016 · version 2
Claimed MSC
11J20
arXiv categories
math.NT, math.HO
Comments
In Russian, published in "Istoriko-matematicheskie Issledovaniya". This version corrects misprints
Journal reference
Istoriko-matematicheskie issledovaniya, 15(50), s.55-64, 2014

Abstract

We discuss Khintchine's theorem on regular matrices (1948) and its exposition in Cassels' book (1957). The paper is written in Russian.

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 quantitative uniformization theorem and the equivalence of the two Chebyshev quantifier formulations are correct.

Theorem 1Correct

One hard inhomogeneous vector controls all target vectors

Pages 6-7 · Theorem 1 and proof scheme · arXiv:1212.5662v2

For η∈NΘ\eta\in N_\Theta, identity (9) converts a good approximation to η\eta into a lower bound for the dual best-approximation product. Mahler transference and Minkowski's successive minima then produce a translated lattice point for every α\alpha with a bound depending only on Δ(Θ,η)\Delta(\Theta,\eta) and the displayed dimensional constants.

Full paper, version 2 ↗
Theorem 2Correct

The point-dependent and uniform Chebyshev constants are equivalent

Page 6 · Theorem 2 · arXiv:1212.5662v2

The uniform formulation trivially implies the point-dependent one. Conversely, applying the latter to the fixed η∈NΘ\eta\in N_\Theta makes Δ\Delta finite; Theorem 1 then supplies one bound for every α\alpha, which is exactly the exchanged quantifier order.

02Proofs2 reported findingsCorrect

The transference body, successive-minimum estimate, and limiting argument are correct and complete.

Equations (9)-(13)Correct and complete

The transference and volume estimates have compatible constants

Pages 6-7 · equations (9)-(13) · arXiv:1212.5662v2

The choice of ν\nu makes the two terms in (9) balance at r0r_0. This yields the positive lower bound ω\omega for ζνmYν+1n\zeta_\nu^mY_{\nu+1}^n. The transposed body has volume (2K)d(ζνmYν+1n)d−1(2K)^d(\zeta_\nu^mY_{\nu+1}^n)^{d-1}, so Minkowski bounds its last minimum by K−dω1−dK^{-d}\omega^{1-d}.

Final limiting stepCorrect and complete

The produced solutions tend to infinity

Page 7 · final paragraph of the proof of Theorem 1 · arXiv:1212.5662v2

The horizontal bound AA diverges with Yν+1Y_{\nu+1} because the positive lower bound on ζνmYν+1n\zeta_\nu^mY_{\nu+1}^n gives Am≫Yν+1nA^m\gg Y_{\nu+1}^n. Hence the translated points supply an unbounded sequence, so the estimate controls the required liminf rather than only finitely many vectors.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1212.5662v2
Authors listed
Nikolay G. Moshchevitin
Audit date
August 20, 2026
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