MathAudit

Paper metadata

Original submission
April 10, 2014
Claimed MSC
11J31
arXiv categories
math.NT
Comments
in German

Abstract

We give a simple proof of a recent result by Kleinbock and Merrill concerning intrinsic approximations on sphere, in the simplest case of two-dimensional sphere in R3\mathbb{R}^3.

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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
01Statements3 reported findingsCorrect

The rational-approximation results on the circle and two-sphere follow from the continued-fraction, lattice, and stereographic-projection arguments given in the paper.

Satz 1 and Bemerkung 1Correct

The circle approximation constant and its sharpness are verified

Sections 2-3 · Satz 1 and Bemerkung 1 · arXiv:1404.2907v1

Writing a rational point by the standard circle parametrization and choosing suitable continued-fraction convergents gives Euclidean error below (1+ε)/(2Q)(1+\varepsilon)/(\sqrt2 Q) infinitely often. The conjugate golden-ratio pattern used in the remark makes the normalized errors tend to the reciprocal constant, so a uniformly larger denominator than 2\sqrt2 is impossible.

Satz 2 and Satz 3Correct

The congruence-constrained planar approximants satisfy the stated bounds

Sections 2 and 4-5 · Satz 2-3 · arXiv:1404.2907v1

Minkowski's theorem applied to the four-dimensional body gives integers (q,b1,b2,a)(q,b_1,b_2,a) satisfying b12+b22=aqb_1^2+b_2^2=aq and the required size inequalities. With t=T/2t=T/2, these become q≤Tq\leq T and ∑i(qβi−bi)2<4q/T\sum_i(q\beta_i-b_i)^2<4q/T. The volume refinement above 3/π\sqrt{3/\pi} yields infinitely many points for Satz 3.

Satz 4 and Satz 5Correct

The sphere estimates follow under stereographic parametrization

Sections 2 and 6 · Satz 4-5 and equation (1) · arXiv:1404.2907v1

The congruence b12+b22≡0(modq)b_1^2+b_2^2\equiv0\pmod q makes the stereographic image a rational point with denominator dividing q2+b12+b22q^2+b_1^2+b_2^2. The displayed Lipschitz comparison between the parameter and sphere distances converts Satz 2 and Satz 3 into the constants 4+o(1)4+o(1) and 23/π+ε2\sqrt{3/\pi}+\varepsilon.

02Proofs1 reported findingCorrect

The continued-fraction and geometry-of-numbers proof chains are complete at the level needed for the stated approximation bounds.

Four-dimensional body argumentCorrect and complete

The Minkowski volume and scaling calculation closes

Sections 4-5 · bodies Kβt\mathfrak K_\beta^t and proof of Satz 2-3 · arXiv:1404.2907v1

The body has volume strictly exceeding the determinant threshold, so it contains a nonzero lattice point. The quadratic identity forces the modular condition, and the two reciprocal scale bounds imply both the denominator cutoff and the error estimate after the stated substitution.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1404.2907v1
Authors listed
Nikolay Moshchevitin
Audit date
August 20, 2026
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