MathAudit

Paper metadata

Original submission
December 29, 2008
Latest revision
September 26, 2009 · version 3
Claimed MSC
11J13
arXiv categories
math.NT
Comments
Submitted to JNTB, correction of misprints

Abstract

We prove a result on the existence of linear forms of a given Diophantine type.

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Audit summary

Audited against arXiv v3

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
01Statements4 reported findingsContains unsupported statements

The quantitative existence assertion in Theorem 5 and its consequence for an individual linear form are verified by the nested half-ball construction. The claimed continuum of such forms is not established: the printed construction fixes its initial data and makes a determined successor choice, producing one limit for each of only two initial half-balls. The uncountability clause of Theorem 4 therefore inherits an unresolved multiplicity obligation. The pointwise inequalities and constants remain correct.

Theorem 5, existence partCorrect

A linear form with the prescribed best-approximation order exists

Pages 5–11 · Theorem 5 and proof · arXiv:0812.4896v3

For every non-increasing ψ\psi with ψ(1)≤(9γ)−1\psi(1)\leq(9\gamma)^{-1}, the induction produces increasing best-approximation vectors mkm_k and nested half-balls Ωk\Omega_k. Their common limit α\alpha satisfies ψ(∣mk∣)−4γψ(∣mk∣)2<∥⟨α,mk⟩∥∣mk∣2≤ψ(∣mk∣)+γψ(∣mk∣)2\psi(|m_k|)-4\gamma\psi(|m_k|)^2<\lVert\langle\alpha,m_k\rangle\rVert |m_k|^2\leq\psi(|m_k|)+\gamma\psi(|m_k|)^2 for every kk. The determinant and radius bounds in Lemma 3 give precisely the two error terms in this display.

Theorem 5, multiplicity clauseNot able to verify

The claimed continuum of forms is not obtained by the proof

Page 5 and pages 10–11 · final sentence of Theorem 5 and conclusion of its proof · arXiv:0812.4896v3

The proof fixes m1,m2,α1m_1,m_2,\alpha_1, permits only the two initial half-balls, and then defines mk+2m_{k+2} by successive minimality conditions, defines αk+1\alpha_{k+1} by two equalities, and selects the unique half-ball nearest the preceding support. It concludes only that the resulting nested sequence has a common point α\alpha. No infinitely branching family, free parameter, or injection from a continuum-sized set is supplied. Integer translations produce only countably many forms. A Cantor-type branching repair may be possible, but it is neither stated nor verified here.

Theorem 4, quantitative inequalitiesCorrect

The individual-form consequence of Theorem 5 is valid

Pages 4–5 · Theorem 4 and its reduction to Theorem 5 · arXiv:0812.4896v3

Choose A=(9γ)−1A=(9\gamma)^{-1} and B=4γB=4\gamma. At a best approximation mkm_k, Theorem 5 gives both required sides. If ∣mk∣≤∣x∣<∣mk+1∣|m_k|\leq|x|<|m_{k+1}|, best-approximation minimality and monotonicity of ψ\psi give the lower bound for xx from that for mkm_k. The infinitely many mkm_k give the upper inequality infinitely often. Thus the quantitative property holds for each form furnished by Theorem 5.

Theorem 4, uncountabilityNot able to verify

The uncountable-set conclusion depends on the unproved multiplicity clause

Page 4 · Theorem 4 · arXiv:0812.4896v3

The text calls Theorem 4 a corollary of Theorem 5. Its assertion of an uncountable set therefore requires the final continuum clause of Theorem 5, not merely the verified construction of one form. Because the proof does not establish that clause, it does not establish uncountability either. The displayed occurrence of ⟨αα,x⟩\langle\alpha\alpha,x\rangle in the lower inequality is separately a harmless duplicated-symbol typo for ⟨α,x⟩\langle\alpha,x\rangle.

02Proofs4 reported findingsContains incorrect or incomplete proofs

The geometric induction correctly proves existence and the stated two-sided estimates after a uniquely determined sign correction in its initial vector. It does not prove that continuum many distinct limit forms arise, although that is a central part of Theorem 5 and is needed for Theorem 4's uncountability conclusion.

Lemma 3Correct

The basis switch and half-ball containment close the induction step

Pages 7–10 · Lemma 3 and proof · arXiv:0812.4896v3

The affine-lattice choice of mk+2m_{k+2} gives the required norm interval and determinant interval. Lemma 2 transfers the integral-level lattice from span⁡Z(mk,mk+1)\operatorname{span}_{\mathbb Z}(m_k,m_{k+1}) to span⁡Z(mk+1,mk+2)\operatorname{span}_{\mathbb Z}(m_{k+1},m_{k+2}). The distance formula for the new center, together with the radius estimate and the definition of γ\gamma, puts the new ball strictly inside the old one. Selecting the half nearer the old support makes mkm_k uniquely best below ∣mk+1∣|m_{k+1}|, while the divisibility corollary handles vectors outside the new lattice.

Induction baseTypo

The first vector needs the opposite sign

Page 11 · displayed definition of m1,m2,α1m_1,m_2,\alpha_1 · arXiv:0812.4896v3

Lemma 3 assumes ⟨mk,mk+1⟩≤0\langle m_k,m_{k+1}\rangle\leq0, but the printed base has m1=(1,0)m_1=(1,0) and a positive first coordinate for m2m_2, so the text's claim that assumption 1 is satisfied is false. Replace that first coordinate by its negative: m2=(−⌈(2γψ1)−1−γ2⌉,3).m_2=\left(-\left\lceil\sqrt{(2\gamma\psi_1)^{-1}-\gamma^2}\right\rceil,3\right). Norms, determinants, the integral-level lattice generated with m1m_1, and all later estimates are unchanged, while the scalar product becomes nonpositive. This is a uniquely repairable sign typo.

Final estimateCorrect

Determinant control and radius control give the two error margins

Page 11 · conclusion of the proof of Theorem 5 · arXiv:0812.4896v3

Lemma 3 gives ψk−1≤∣det⁡(mk+1,mk+2)∣∣mk∣2<ψk−1+3γ<(ψk−3γψk2)−1.\psi_k^{-1}\leq\frac{|\det(m_{k+1},m_{k+2})|}{|m_k|^2}<\psi_k^{-1}+3\gamma<(\psi_k-3\gamma\psi_k^2)^{-1}. The new half-ball has Rk+1∣mk∣3≤γψk2R_{k+1}|m_k|^3\leq\gamma\psi_k^2. The first relation locates the center value and the second bounds its variation throughout Ωk+1\Omega_{k+1}, yielding the lower margin 4γψk24\gamma\psi_k^2 and upper margin γψk2\gamma\psi_k^2 exactly as stated.

Continuum conclusionIncomplete

No branching argument is supplied

Pages 10–11 · iteration and final sentence of the proof · arXiv:0812.4896v3

The iteration proves the existence of a nested sequence and one common point. It neither exhibits infinitely many stages with two admissible successors nor varies the initial data over a continuum while preserving the induction hypotheses. The phrase 'which proves Theorem 5' therefore skips the proof obligation created by 'Moreover, there is a continuum of such α\alpha.' Because the next theorem uses that multiplicity, this is not merely an editorial omission.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:0812.4896v3
Authors listed
Oleg N. German, Nikolay G. Moshchevitin
Audit date
August 20, 2026
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