MathAudit

arXiv:0804.0120v3

Proof of W.M.Schmidt's conjecture concerning successive minima of a lattice

Nikolay G. Moshchevitin

math.NT11H0611J13
View on arXiv ↗Original PDF ↗

Paper metadata

Original submission
April 1, 2008
Latest revision
December 8, 2010 · version 3
Claimed MSC
11H06, 11J13
arXiv categories
math.NT
Comments
Submitted to Proceedings of LMS, further minor corrections

Abstract

We prove W.M. Schmidt's conjecture about a one-parameter family of lattices related to simultaneous Diophantine approximations.

AI-generated audit

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
01Statements2 reported findingsContains unsupported statements

The case k=1k=1 is verified, but the general case 2≤k≤n−12\leq k\leq n-1 depends on a low-height cylinder exclusion in Lemma 7 that the printed hypotheses and argument do not establish.

Theorem 1, $k=1$Correct

The two-dimensional rational-plane construction

Pages 2–6 · Theorem 1 and Section 2 · arXiv:0804.0120v3

The rational approximants are chosen in successive complete two-dimensional sublattices. Lemmas A and B control the first and third minima on complementary compact ranges, while the nested balls avoid every rational hyperplane. Their unique limit therefore has the required independence and limiting minima for k=1k=1.

Theorem 1, general caseNot able to verify

The construction for k≥2k\geq2 is not established by Lemma 7

Pages 16–23 · Lemma 7 and the general-case induction · arXiv:0804.0120v3

The induction uses Lemma 7 to produce an empty cylinder in the enlarged lattice. In the range ∣q∣≤H|q|\leq H, the proof needs to pass from closeness to the shifted center qξ′q\xi' to the (Λ,γ,W)(\Lambda,\gamma,W)-bad lower bound centered at qξq\xi. The printed inclusion supplies no margin for the center displacement, and the issue is not an immediate consequence of bad approximability. Shrinking the auxiliary constant may plausibly provide such a margin, but the resulting changes must be propagated through Lemmas 8 and 5 and the final induction; no fully verified repair of all those dependencies is supplied here.

02Proofs2 reported findingsContains incorrect or incomplete proofs

Lemma 7 contains a concrete unproved cylinder inclusion used throughout the general induction. Several additional dimension and parameter symbols are typographical only.

Lemma 7Incomplete as written · plausible repair only

The low-height shifted cylinder is not shown to lie in the bad-approximation cylinder

Pages 15–16 · final paragraph of the proof of Lemma 7 · arXiv:0804.0120v3

For 0≤∣q∣≤H0\leq |q|\leq H, the proof asserts that Cξ′(H,γ′T−1/k)∩Λ\mathcal C_{\xi'}(H,\gamma'T^{-1/k})\cap\Lambda is contained in Cξ(H,γH−1/(k−1))∩Λ\mathcal C_\xi(H,\gamma H^{-1/(k-1)})\cap\Lambda. Its own definitions give γ′T−1/k=γH−1/(k−1)\gamma'T^{-1/k}=\gamma H^{-1/(k-1)}, while the two center lines are distinct by an amount that can reach the same scale at ∣q∣=H|q|=H. Thus the equality of radii leaves no triangle-inequality margin, and the asserted inclusion does not follow. A smaller γ′\gamma' and a correspondingly rescaled HH appear capable of creating a strict margin, but every later constant depending on Lemma 7 would have to be recomputed.

General-case notationTypo

Several ambient dimensions and parameters have unique corrections

Pages 3 and 18–23 · cylinder identification and final induction · arXiv:0804.0120v3

Replace ZN+1\mathbb Z^{N+1} by Zn+1\mathbb Z^{n+1} in the cylinder identification, use q∈Zn+1q\in\mathbb Z^{n+1} for a vector orthogonal to a subspace of Rn+1\mathbb R^{n+1}, restore the missing minus sign in Tt−1/kT_t^{-1/k}, and replace the undefined Tt+1T_{t+1} in the new-stage badness parameter by Wt+1=TtW_{t+1}=T_t. Each correction is fixed by the definitions immediately surrounding it.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

Detailed audit reportFull reasoning, manuscript locations, and references.
Open report PDF ↗

Author response

Challenge an audit finding

Local workflow preview

A listed author may submit formal evidence that an audit is inaccurate. The response would be considered in a fresh AI re-evaluation; it would not edit the audit automatically.

Paper
arXiv:0804.0120v3
Authors listed
Nikolay G. Moshchevitin
Audit date
August 20, 2026
  1. 01Establish identityMatch an authenticated scholarly identity to this paper.
  2. 02Submit evidenceIdentify the finding and give a formal mathematical response.
  3. 03Re-evaluateA separate agent checks the response and records a disposition.
Recommended production method

Authenticate with ORCID, then require an exact arXiv match

MathAudit should accept the identity only when ORCID OAuth authenticates the claimant's iD and this exact arXiv paper appears in arXiv's public authority feed for that iD. A matching name alone is not sufficient.

ORCID OAuth and arXiv authority-record lookup are not connected in this local prototype.

Email fallback for papers without a linked ORCID

A production fallback could send a one-time link only when the submitted address matches an independently maintained author-contact allowlist for this paper. MathAudit must return the same message for every address so the form cannot reveal which contacts are on that list.

This demonstration does not send, store, or compare the address.

Structured response preview

This form remains unavailable until production identity verification succeeds. Nothing entered here is submitted.

This panel never establishes authorship in the local prototype. A production result should be described narrowly as an authenticated ORCID match or control of a separately allowlisted author-contact mailbox.