MathAudit

Paper metadata

Original submission
September 8, 2012
Claimed MSC
11J13
arXiv categories
math.NT
Comments
15 pages

Abstract

We improve on Jarník's inequality between the uniform Diophantine exponent αα and the ordinary Diophantine exponent ββ for a system of n≥2n\geq 2 real linear forms in two integer variables. Jarník proved that β≥α(α−1)β\geq α(α-1). We give a better bound for α>1α>1, proving β≥{12(α2−α+1+(α2−α+1)2+4α2(α−1)),1≤α≤2,12(α2−1+(α2−1)2+4α(α−1)),α≥2.β\geq\begin{cases}\frac12\left(α^2-α+1+\sqrt{(α^2-α+1)^2+4α^2(α-1)}\right),&1\leq α\leq 2,\\\frac12\left(α^2-1+\sqrt{(α^2-1)^2+4α(α-1)}\right),&α\geq 2.\end{cases}

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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 improved lower bound for the ordinary exponent of two linear forms is correct under the stated rank hypothesis.

Theorem 1Correct

Improved exponent bound in the rank-at-least-four case

Pages 4 and 13-14 · Theorem 1 and Section 8 · arXiv:1209.1697v1

The proof produces either Zν+1≫Zνg(α)Z_{\nu+1}\gg Z_\nu^{g(\alpha)} or Zk+1≫Zkg(α)Z_{k+1}\gg Z_k^{g(\alpha)} on infinitely many admissible blocks. Substituting the uniform error estimates gives β(Θ)≥αg(α)\beta(\Theta)\geq\alpha g(\alpha), and balancing the two alternatives yields equation (13), hence G(α)G(\alpha).

Full paper, version 1 ↗
Rank hypothesisCorrect

The four-vector assumption is used exactly where required

Pages 4 and 11-13 · Remark 2 and Lemmas 4-6 · arXiv:1209.1697v1

The hypothesis ensures that the eventual span of best approximations has enough independent coordinate columns to form the nonzero determinants in Lemmas 5-6. The duplicated-row example in Remark 2 shows why this cannot be omitted from the stated theorem.

02Proofs3 reported findingsCorrect

The geometric two-plane estimate, determinant growth lemmas, and final optimization are correct; one squared-distance symbol is a harmless typo.

Lemma 2Typo · no status impact

One coordinate is repeated in the distance formula

Page 7 · proof of Lemma 2 · arXiv:1209.1697v1

For L={η1=η2=0}L=\{\eta_1=\eta_2=0\}, the distance is η12+η22\sqrt{\eta_1^2+\eta_2^2}, not η12+η12\sqrt{\eta_1^2+\eta_1^2}. The following expansion uses both p1,p2p_1,p_2 and q1,q2q_1,q_2, so the correction is mechanically determined and leaves the monotonicity proof unchanged.

Lemma 3Correct and complete

The planar ellipse comparison is valid

Pages 9-10 · Lemma 3 and Corollary · arXiv:1209.1697v1

Minkowski bounds the product of the minor semiaxis and the next height at both ends of the block. The angle exclusion gives ξl≥aΞl\xi_l\geq a\Xi_l, and central projection plus Lemma 2 compares the endpoint ratios. Combining them yields (26) and then (27).

Lemmas 5-6Correct and complete

The nonzero determinant bounds give the required growth

Pages 12-13 · Lemmas 5-6 · arXiv:1209.1697v1

The selected coordinate columns form nonzero integral determinants. Replacing the yy columns by their errors produces the upper bounds Zν−αZk1−αZk+1Z_\nu^{-\alpha}Z_k^{1-\alpha}Z_{k+1} and Zν1−α(Θ)+εZν+1Z_\nu^{1-\alpha(\Theta)+\varepsilon}Z_{\nu+1}, respectively, which rearrange to the printed growth estimates.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1209.1697v1
Authors listed
Nikolay G. Moshchevitin
Audit date
August 20, 2026
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