MathAudit

Paper metadata

Original submission
March 3, 2012
Claimed MSC
11J25, 11J82, 11H06
arXiv categories
math.NT

Abstract

In this paper we give a simple proof of an inequality for intermediate Diophantine exponents obtained recently by W. M. Schmidt and L. Summerer.

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
01Statements2 reported findingsCorrect

Both Schmidt-Summerer inequalities are correct; the local lattice lemma contains a harmless systematic index typo.

Theorem 1Correct

Lower and upper intermediate-exponent inequalities

Page 3 and Sections 4.3-4.5 · Theorem 1 · arXiv:1203.0641v1

The front-facet reduction gives paired inequalities for the first and ppth successive minima. Taking liminf and limsup yields (5). Applying the same argument to the dual lattice and path, then Proposition 4, changes pp to d+1−pd+1-p and gives (6) with the correct signs.

Full paper, version 1 ↗
Propositions 1-2Correct

The exponent conversion and endpoint bounds are consistent

Page 2 · Propositions 1-2 · arXiv:1203.0641v1

Solving (1+βp)(1+ψ‾p)=d/n(1+\beta_p)(1+\underline\psi_p)=d/n and its uniform analogue reverses order exactly once because the denominator is positive. Thus βp≥αp≥0\beta_p\geq\alpha_p\geq0 becomes −1≤ψ‾p≤ψ‾p≤m/n-1\leq\underline\psi_p\leq\overline\psi_p\leq m/n, as stated.

02Proofs2 reported findingsCorrect

The geometric lemma, front-facet reduction, asymptotic passage, and transference are correct after a uniquely determined index correction.

Lemma 1Typo · no status impact

Coordinate indices in the reduction step are typographical errors

Pages 3-4 · proof of Lemma 1 · arXiv:1203.0641v1

After defining vij′=vij−⌊vi1/v1⌋vjv'_{ij}=v_{ij}-\lfloor v_{i1}/v_1\rfloor v_j, the range must be j=2,…,dj=2,\ldots,d and the bound must end in 2λhj2\lambda h_j, not j=2,…,p−1j=2,\ldots,p-1 and 2λhi2\lambda h_i. These are forced coordinate substitutions; with them every vi′v'_i lies in 2P32P_3 and independence is unchanged.

Sections 4.3-4.5Correct and complete

The limiting and duality arguments close

Pages 5-7 · equations (13)-(26) · arXiv:1203.0641v1

The irrationality hypothesis makes both relevant side lengths diverge in the required directions, so the additive log⁡2\log2 errors vanish after division by s0s_0. Proposition 3 permits the liminf subsequence, and the exact duality formulas in Proposition 4 transform (25) into (6) without changing a liminf into a limsup.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1203.0641v1
Authors listed
Oleg N. German, Nikolay G. Moshchevitin
Audit date
August 20, 2026
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  2. 02Submit evidenceIdentify the finding and give a formal mathematical response.
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