arXiv:0904.1906v2

On Diophantine approximations with positive integers: a remark to W.M.Schmidt's theorem

Nikolay G. Moshchevitin

math.NT11J13

Paper metadata

Original submission: April 13, 2009
Latest revision: December 21, 2011 · version 2
Claimed MSC: 11J13
arXiv categories: math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: correction of typos

Abstract

We prove a generalization of W.M. Schmidt's theorem related to Diophantine approximations for a linear form of the type $α_1x_1+α_2x_2+y$ with positive integers $x_1,x_2$ .

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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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Detailed mathematical audit

Generated August 20, 2026

01Statements2 reported findingsCorrect

The quantitative positive-coordinate approximation theorem and its local best-approximation alternative are correct; a verified repair is needed in the printed proof of Lemma 1.

Theorem 2Correct

Positive-coordinate approximation under a Diophantine lower bound

Pages 1–3 · Theorem 2 and its reduction to Theorem 3 · arXiv:0904.1906v2

The alternatives of Theorem 3 provide infinitely many positive integer pairs. Combining $\Gamma M_\nu^{-\gamma}\leq M_{\nu+1}^{-2}$ with alternative (ii) gives the displayed lower bound for $M_\nu$ in terms of the positive solution size; substitution into its error estimate yields exactly $g(\gamma)=\tau+(2\tau-2)/(\tau^2\gamma-2)$ and the stated constant $C(\Gamma)$ .

Theorem 3Correct

Local alternative for three independent best approximations

Page 3 · Theorem 3 · arXiv:0904.1906v2

Corollaries 1–3 exhaust the cases determined by $\zeta_{\nu+1}$ and $\zeta_\nu$ . Their size bounds imply the common upper bound in alternative (ii), while the golden-ratio identity $\tau-1=1/\tau$ gives the stated exponent. The repair to Lemma 1 below supplies its required positive point without changing any constant.

02Proofs4 reported findingsContains incorrect or incomplete proofs

Lemma 1 makes an invalid equality-boundary inference as printed. A verified local perturbation repairs it and preserves all theorem statements and constants.

Lemma 1Incomplete as written · verified repair

Minkowski at the critical volume does not itself give strict inequalities

Page 4 · proof of Lemma 1 · arXiv:0904.1906v2

The closed parallelepiped has volume exactly $8$ , so Minkowski supplies a nonzero lattice point only with non-strict boundary inequalities. Linear independence excludes a zero linear-form value, but it does not exclude $|\alpha_1x_1+\alpha_2x_2+y|=\zeta_\nu$ , nor does it by itself force both $x_1,x_2$ to be nonzero. Thus the printed sentence claiming strict positivity and strict error directly from linear independence is not valid. Verified repair: for $0<\varepsilon<1$ , replace the error and difference bounds by $(1-\varepsilon)\zeta_\nu$ and $(1-\varepsilon)M_{\nu+1}$ , and replace $R_\nu$ by $R_\nu/(1-\varepsilon)^2$ ; the volume remains $8$ . The resulting point has strict error, hence lies outside the earlier best-approximation body and has $\max|x_i|\geq M_{\nu+1}$ ; the strict difference bound forces the two coordinates to have the same nonzero sign. Taking a sequence $\varepsilon\downarrow0$ , boundedness gives one repeated lattice point, which has $\max|x_i|\leq R_\nu$ and the same strict error and sign properties. Negating it if necessary proves Lemma 1 with every printed constant.

Lemma 1 notationTypo

Two subscripts name the wrong bodies

Page 4 · proof of Lemma 1 · arXiv:0904.1906v2

Replace $R_\mu$ by $R_\nu$ in the volume identity and replace $\Omega_2\setminus\Omega_1$ by $\Omega_\nu^1\setminus\Omega_\nu$ . The definitions on the same page uniquely determine both corrections.

Lemma 2Correct and complete

Lattice-basis and coefficient bounds

Pages 5–6 · Lemma 2 and Equations (6)–(8) · arXiv:0904.1906v2

The nonzero integer determinant and the two small-error hypotheses give $D_\nu\geq M_\nu^2/2$ . Minkowski's second-minimum argument in the lattice generated by $\xi_\nu,\xi_{\nu+1}$ produces an independent point within the thin rectangle; its perpendicular coefficient is one in absolute value, so it and $\xi_\nu$ form a lattice basis. The translated fundamental-domain bound then gives the positive point and the displayed coefficient and error estimates.

Proof of Theorem 3Correct and complete

The three corollaries cover every case

Pages 6–7 · proof of Theorem 3 · arXiv:0904.1906v2

If $\zeta_{\nu+1}\geq(8M_{\nu+2}^2)^{-1}$ , Corollary 1 gives alternative (i). Otherwise the second hypothesis of Lemma 2 holds, and the two complementary ranges of $\zeta_\nu$ invoke Corollary 2 or Corollary 3. Their bounds imply alternative (ii) because $M_{\nu+1}^{\tau-1}\geq M_\nu^{1/\tau}$ .

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:0904.1906v2
Authors listed: Nikolay G. Moshchevitin
Audit date: August 20, 2026

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On Diophantine approximations with positive integers: a remark to W.M.Schmidt's theorem

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Positive-coordinate approximation under a Diophantine lower bound

Local alternative for three independent best approximations

Minkowski at the critical volume does not itself give strict inequalities

Two subscripts name the wrong bodies

Lattice-basis and coefficient bounds

The three corollaries cover every case

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