MathAudit

Paper metadata

Original submission
August 22, 2011
Claimed MSC
11J13
arXiv categories
math.NT
Comments
8 pages

Abstract

We show that there exist real numbers α1,α2α_1,α_2 linearly independent over Z\mathbb{Z} together with 1 such that for every non-zero integer vector (m1,m2)(m_1,m_2) with m1≥0m_1\ge 0 and m2≥0m_2\ge 0 one has ∣∣m1α1+m2α2∣∣≥2−300(max⁡(m1,m2))−σ||m_1α_1+m_2α_2|| \ge 2^{-300} (\max(m_1, m_2))^{-σ} with σ=1.94696+σ= 1.94696^+.

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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 findingsContains unsupported statements

The reduction from the Fundamental Lemma to the counterexample is correct, but the paper does not establish the Fundamental Lemma's decisive lattice-selection and linear-independence assertions.

Theorem 1Not able to verify

The counterexample depends on an unverified construction lemma

Pages 1-3 and 5-8 · Theorem 1, Fundamental Lemma, and Sections 5-6 · arXiv:1108.4435v1

Lemmas 1-2 would give the stated 2−300M−σ2^{-300}M^{-\sigma} lower bound if the sequence in the Fundamental Lemma existed. Section 6, however, only calls the candidate lattice points dense at scale o(Hν)o(H_\nu) and then asserts that one lies in the prescribed disk while simultaneously satisfying completeness, sign, angle, nesting, and eventual rational-independence conditions. No quantitative covering or avoidance argument proves those simultaneous requirements, so the central existence claim remains unsupported.

Full paper, version 1 ↗
Fundamental LemmaNot able to verify

The five asserted properties are not obtained by the written induction

Pages 2-3 and 6-8 · Fundamental Lemma and its proof sketch · arXiv:1108.4435v1

The construction must produce a primitive mν+1m_{\nu+1} in DνD_\nu, make ⟨mν,mν+1⟩Z\langle m_\nu,m_{\nu+1}\rangle_{\mathbb Z} complete, retain both angle exclusions, and leave a choice avoiding every rational relation for the limiting pair. The final two paragraphs assert these properties from having 'many points' but give neither a lower count in DνD_\nu nor an upper count for the excluded subsets. These are nontrivial obligations, not immediate consequences of the preceding estimates.

02Proofs4 reported findingsContains incorrect or incomplete proofs

The determinant and dependent-lattice estimates are correct, but the proof of the construction lemma is incomplete at the step on which the theorem depends.

Lemma 1Correct and complete

The independent-vector determinant estimate is correct

Page 4 · Lemma 1 · arXiv:1108.4435v1

The nonzero integer determinant is split into one term containing ζ(m)\zeta(m) and two terms containing ζν,ζν+1\zeta_\nu,\zeta_{\nu+1}. The upper endpoint of IνI_\nu makes the latter contribution at most one half, and the lower endpoint converts the remaining determinant bound into ∣ζ(m)∣≥M−σ|\zeta(m)|\geq M^{-\sigma}.

Lemma 2Correct and complete

The dependent-vector lattice estimate is correct

Pages 4-5 · Lemma 2 · arXiv:1108.4435v1

Completeness gives integral coefficients in m=λmν+μmν+1m=\lambda m_\nu+\mu m_{\nu+1}. The sign and angle hypotheses bound both MM and ∣λ∣|\lambda| below in terms of ∣μ∣|\mu|, while the two-sided estimates for ζν\zeta_\nu prevent cancellation. Eliminating Mν+1M_{\nu+1} gives the printed exponent σ\sigma and constant.

Section 6Incomplete as written · no repair supplied

The lattice-point selection and irrationality steps are incomplete

Pages 7-8 · final four paragraphs of the Fundamental Lemma sketch · arXiv:1108.4435v1

The estimate Mνμ∗=o(Hν)M_\nu\mu_*=o(H_\nu) controls a mesh scale but does not by itself prove that the constrained affine lattice meets the fixed-radius disk DνD_\nu, much less that enough admissible points remain to impose (iv), (v), and avoid countably many rational hyperplanes in the limit. A repair requires explicit lattice-covering and branching estimates throughout the induction; none are supplied.

Local notationTypos · no status impact

Four symbols in the construction and dependent-vector estimate are typographical

Pages 2, 5, and 7 · condition (iv), equation (o3), and Section 6 lattice definition · arXiv:1108.4435v1

Condition (iv) labels both standard basis vectors e1e_1, so the second is e2e_2; the coefficient bound in Lemma 2 has MμM_\mu where the surrounding lines require MνM_\nu; the first power of Mν+1M_{\nu+1} in (o3) is missing its minus sign; and the affine-lattice definition repeats λ1\lambda_1 where the coefficient of mνm_\nu must be λ2\lambda_2. All four repairs are forced locally and do not address the separate construction gap.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1108.4435v1
Authors listed
Nikolay G. Moshchevitin
Audit date
August 20, 2026
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