MathAudit

Paper metadata

Original submission
December 12, 2008
Claimed MSC
11J13
arXiv categories
math.NT
Comments
Submitted to the Proceedings of the Department of Mathematics and Mechanics, MSU

Abstract

We prove a new quantitative result on the degeneracy of the dimension of the subspace spanned by the best Diophantine approximations for a linear form.

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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
01Statements2 reported findingsCorrect

The convergence criterion does force almost every extension to have eventually exactly the embedded original best approximations.

Theorem 1Correct

Almost-everywhere stabilization of best approximations

Pages 2–5 · Theorem 1 and its proof · arXiv:0812.2455v1

The exceptional sets are bounded by the stated convergent series, so Borel–Cantelli excludes all but finitely many competing extensions. The exponential growth lemma for best-approximation denominators makes the remaining comparison uniform, and the embedded original approximations therefore exhaust the eventual best-approximation sequence for almost every added parameter.

Theorem 2 and Corollary 1Correct

Extensions of ψ\psi-singular linear forms

Pages 2–3 · Theorem 2 and Corollary 1 · arXiv:0812.2455v1

For a ψ\psi-singular rr-tuple, the defining inequality gives ζν≤ψ(Mν+1)\zeta_\nu\leq\psi(M_{\nu+1}). Substitution in Theorem 1 yields the series in Theorem 2. Lemma 1 makes MνM_\nu grow exponentially, so the logarithmically strengthened choice of ψ\psi in Corollary 1 makes that series converge and verifies the stated almost-everywhere stabilization.

02Proofs2 reported findingsCorrect

The counting, Borel–Cantelli, and best-approximation comparison arguments are complete. The displayed dimension indices have harmless unique corrections.

Lemma 1Correct

Best-approximation heights grow exponentially

Pages 3 and 5 · Lemma 1 and its proof · arXiv:0812.2455v1

Among a sufficiently long block of best approximations, pigeonhole reduction modulo two produces two vectors in the same parity class. Their half-difference is an integer vector of smaller height; best-approximation minimality then forces the claimed doubling over a bounded number of indices. Iteration gives precisely the exponential growth used in the convergence argument.

Notation in Section 1Typo

Three occurrences use the ambient dimension index incorrectly

Page 2 · definition of best approximations and MνM_\nu · arXiv:0812.2455v1

The vector (m0,…,mr)(m_0,\ldots,m_r) is first printed as an element of Zr\mathbb Z^r, and two maxima are printed with upper index nn. Replace these by Zr+1\mathbb Z^{r+1} and upper index rr, respectively. The next lines already use those corrected dimensions, so the repairs are mechanical and have no proof effect.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:0812.2455v1
Authors listed
Oleg N. German, Nikolay G. Moshchevitin
Audit date
August 20, 2026
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