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arXiv:1312.1841v1

Über eine Ungleichung von Schmidt und Summerer für diophantische Exponenten von Linearenformen in drei Variablen

Nikolay Moshchevitin

math.NT11J13
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Paper metadata

Original submission
December 6, 2013
Claimed MSC
11J13
arXiv categories
math.NT
Comments
in German

Abstract

We give a simple proof of a recent inequality by W.M. Schmidt and L. Summerer concerning Diophantine exponents for a linear form in three real variables.

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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.

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Generated August 20, 2026
01Statements3 reported findingsCorrect

The advertised Schmidt–Summerer lower bound for the ordinary exponent of one linear form in three variables is correct. The proof reduces the best-approximation sequence to a three-dimensional tail or to recurring four-vector configurations and obtains the stated exponent in both cases. Two introductory notation errors are uniquely repairable typos and do not affect the central result.

Satz 2Correct

The Schmidt–Summerer exponent bound

Pages 1–3 · Satz 2 and its proof · arXiv:1312.1841v1

For every proper triple Θ=(θ1,θ2,θ3)\Theta=(\theta_1,\theta_2,\theta_3), the paper proves ω(Θ)≥ω^(Θ)4ω^(Θ)−3−12.\omega(\Theta)\geq\widehat\omega(\Theta)\frac{\sqrt{4\widehat\omega(\Theta)-3}-1}{2}. Taking any α<ω^\alpha<\widehat\omega, the best-approximation estimate Lj−1≤Mj−αL_{j-1}\leq M_j^{-\alpha} holds eventually. If the tail lies in a three-dimensional subspace, the cited two-dimensional Jarník argument gives the stronger bound. Otherwise the recurring four-vector configuration supplies one of two growth alternatives, each yielding Lj≪Mj−αλL_j\ll M_j^{-\alpha\lambda} with λ=(4α−3−1)/2\lambda=(\sqrt{4\alpha-3}-1)/2. Letting α\alpha increase to ω^\widehat\omega gives exactly the displayed statement.

Best-approximation dichotomyCorrect

The two geometric cases cover the sequence

Page 2 · Fall 1 and Fall 2 · arXiv:1312.1841v1

If all sufficiently late best-approximation vectors lie in one three-dimensional real subspace, the argument reduces to the previously established three-coordinate estimate cited as the stronger inequality (2). If this does not happen, the standard maximal two-plane blocks of consecutive best approximations yield infinitely many indices ν<k\nu<k with the four independence and incidence properties listed in Fall 2. Thus no third asymptotic configuration is omitted.

Introductory Jarník formulaTypo

The sign in the two-variable background inequality is mistyped

Page 1 · Satz 1, equation (2) · arXiv:1312.1841v1

The cited Jarník inequality for one linear form in two variables is ω≥ω^(ω^−1)\omega\geq\widehat\omega(\widehat\omega-1), not the printed expression with ω^+1\widehat\omega+1. The minus sign is also the form used by the related papers cited in the bibliography. This is a background citation and is not used in the proof of Satz 2 except through the correct stronger three-dimensional-tail result.

02Proofs3 reported findingsCorrect

The proof of the central inequality is correct and complete at the stated level of reliance on the cited best-approximation geometry. Independence gives the determinant estimate, the common two-plane gives the covolume comparison, and the chosen interpolation parameters force one of two estimates with the same final exponent. The remaining issues are harmless notation typos.

Equations (5) and (6)Correct

Plane covolumes and four-vector independence provide the central product bound

Page 2 · equations (5)–(6) · arXiv:1312.1841v1

Within a maximal block in the same rational two-plane, the consecutive best-approximation parallelograms have comparable nonzero covolume, giving LνMν+1≍Lk−1MkL_\nu M_{\nu+1}\asymp L_{k-1}M_k. The four independent integer vectors mν−1,mν,mν+1,mk+1m_{\nu-1},m_\nu,m_{\nu+1},m_{k+1} have a nonzero integral determinant. Expanding that determinant with one linear-form coordinate and three height coordinates gives 1≪Lν−1MνMν+1Mk+1.1\ll L_{\nu-1}M_\nu M_{\nu+1}M_{k+1}. Combining these two relations is the valid starting point for the interpolation step.

Equations (7)–(10)Correct

Both interpolation alternatives give the same exponent

Pages 2–3 · equations (7)–(10) · arXiv:1312.1841v1

With b=−1+4α−32(α−1),a=1−b,b=\frac{-1+\sqrt{4\alpha-3}}{2(\alpha-1)},\qquad a=1-b, one has a,b∈[0,1]a,b\in[0,1] and λ=b(α−1)\lambda=b(\alpha-1). Splitting the combined product inequality at powers aa and bb forces either (7) or (8). In the first case Mk+1≫MkλM_{k+1}\gg M_k^\lambda and hence Lk≪Mk−αλL_k\ll M_k^{-\alpha\lambda}; in the second, substituting Mν+1≤Lν−1/αM_{\nu+1}\leq L_\nu^{-1/\alpha} gives Lν≪Mν−αλL_\nu\ll M_\nu^{-\alpha\lambda}. Therefore infinitely many best approximations attain exponent at least αλ\alpha\lambda, as required.

Definition of $\psi_\Theta$Typo

The general-nn definition accidentally lists only three integer variables

Page 1 · opening definition of ψΘ\psi_\Theta · arXiv:1312.1841v1

The paragraph begins with arbitrary nn, so the minimum must range over x1,…,xn∈Zx_1,\ldots,x_n\in\mathbb Z, not only x1,x2,x3x_1,x_2,x_3. The displayed linear form already uses the corresponding nn coefficients conceptually, and the paper immediately specializes the main argument to n=3n=3. Replacing the variable list is a unique notation repair.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1312.1841v1
Authors listed
Nikolay Moshchevitin
Audit date
August 20, 2026
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