MathAudit

Paper metadata

Original submission
February 10, 2019
Latest revision
May 15, 2019 · version 5
Claimed MSC
11J13
arXiv categories
math.NT
Comments
in German, this version contains minor corrections

Abstract

We prove a metric statement about approximation of a nn-dimensional linear subspace AA in Rd\mathbb{R}^d by nn-dimensional rational subspaces. We consider the problem of finding a rational subspace BB of bounded height H=H(B)H=H(B) for which the angle of inclination ψ(A,B)ψ(A,B) is small in terms of HH. In the simplest case d=4,n=2d=4, n=2 we give a partial solution of a problem formulated by W.M. Schmidt in 1967.

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Audit summary

Audited against arXiv v5

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 metric theorem for two-dimensional subspaces of R4\mathbb R^4 is correctly proved. The general even-dimensional Theorem 2 is explicitly stated without proof and could not be independently verified from this manuscript.

Theorem 1Correct

The almost-everywhere angle lower bound is correct

Pages 3–12 · Theorem 1 and Sections 2–6 · arXiv:1902.03546v5

Rational planes are counted by height, their small-angle neighborhoods have the asserted measure bound, and the convergence hypothesis makes the exceptional limsup summable. Borel–Cantelli then yields the claimed uniform lower bound for almost every target plane.

Full paper, version 5 ↗
Theorem 2Not able to verify

The even-dimensional generalization is stated without proof

Pages 12–13 · Section 7 and Theorem 2 · arXiv:1902.03546v5

Section 7 says explicitly that the general statement is formulated without proof and only remarks that the argument should be analogous. The higher-dimensional matrix count and neighborhood-volume estimate needed to justify the new series exponent are not supplied or cited as an existing theorem. No counterexample was established, but the claim remains unsupported here.

02Proofs2 reported findingsContains unverified proofs

The proof of Theorem 1 is complete. Theorem 2 has no proof in the submitted version, so the generalization is unverified rather than shown false.

Proof of Theorem 1Correct and complete

The rational-plane count and Borel–Cantelli estimate close

Pages 4–12 · Sections 2–6 and proof of Theorem 1 · arXiv:1902.03546v5

Primitive Plucker coordinates count rational two-planes in each dyadic height shell with the stated power. The coordinate charts on the Grassmannian bound the measure of the small-angle neighborhood of each plane, and summing these bounds gives exactly the convergence series in Theorem 1. Borel–Cantelli then removes only a null limsup set.

Section 7Not supplied

The general theorem omits its counting and measure argument

Pages 12–13 · Theorem 2 and following remarks · arXiv:1902.03546v5

The two-plane proof does not automatically establish the announced d=2sd=2s exponent: one must count primitive rank-ss lattices of bounded height and bound the measure of their angular neighborhoods with the precise power used in the convergence series. Those steps are absent.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1902.03546v5
Authors listed
Nikolay Moshchevitin
Audit date
August 20, 2026
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