MathAudit

Paper metadata

Original submission
January 6, 2013
Latest revision
May 25, 2013 · version 4
Claimed MSC
arXiv categories
math.NT, math.DS
Comments
Incorporated readers' suggestions and clarified exposition

Abstract

We quantify the density of rational points in the unit sphere SnS^n, proving analogues of the classical theorems on the embedding of Qn\mathbb{Q}^n into Rn\mathbb{R}^n. Specifically, we prove a Dirichlet theorem stating that every point α∈Snα\in S^n is sufficiently approximable, the optimality of this approximation via the existence of badly approximable points, and a Khintchine theorem showing that the Lebesgue measure of approximable points is either zero or full depending on the convergence or divergence of a certain sum. These results complement and improve on previous results, particularly recent theorems of Ghosh, Gorodnik and Nevo.

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Audit summary

Audited against arXiv v4

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 19, 2026
01Statements2 reported findingsContains unsupported statements

The Dirichlet theorem on the sphere, the badly approximable thickness result, and the Lebesgue zero-full law are correct. The general Hausdorff-measure zero-full law in Theorem 1.4 is not able to be verified under its printed hypotheses because its proof requires a regularity assumption on the dimension function that the theorem neither states nor derives. The power-function specialization used for the theorem's Hausdorff-dimension formula does satisfy that assumption.

Theorems 1.1–1.3 and 1.5Correct

Intrinsic approximation, thickness, and the Lebesgue zero-full law

Pages 3–4 and 6 · Theorems 1.1–1.3 and 1.5 · arXiv:1301.0989v4

The light-cone dictionary identifies rational points on SnS^n with primitive isotropic lattice data at the stated denominator scale. The flow correspondence turns ϕ\phi-approximability into infinitely many cusp excursions. The upper counting estimate proves the convergence case, while the shrinking-target theorem and the estimate of cusp-neighborhood measure give the divergence case with the series ∑kkn−1ϕ(k)n\sum_k k^{n-1}\phi(k)^n. Dani's bounded-trajectory dimension result supplies the asserted thickness of badly approximable points.

Theorem 1.4Not able to verify

The Hausdorff-measure divergence conclusion lacks a required regularity hypothesis

Pages 4 and 20–21 · Theorem 1.4, Theorem 4.7, and proof of Theorem 1.4 · arXiv:1301.0989v4

Theorem 4.7, the Mass Transference Principle used for the divergence case, explicitly assumes that r−nf(r)r^{-n}f(r) is monotonic. Theorem 1.4 assumes only that ff is a dimension function and that knf(ϕ(k))k^n f(\phi(k)) is non-increasing. The latter does not imply the former: for example, with ϕ(k)=k−2\phi(k)=k^{-2} and f(r)=rn(2+14sin⁡(log⁡r))f(r)=r^n(2+\tfrac14\sin(\log r)), the function ff is increasing and continuous, knf(ϕ(k))=k−n(2−14sin⁡(2log⁡k))k^n f(\phi(k))=k^{-n}(2-\tfrac14\sin(2\log k)) is decreasing, but r−nf(r)=2+14sin⁡(log⁡r)r^{-n}f(r)=2+\tfrac14\sin(\log r) is not monotonic. The proof invokes Theorem 4.7 without establishing its omitted hypothesis. No counterexample to the printed zero-full conclusion itself was found, but no independent proof for arbitrary dimension functions under exactly the printed assumptions was verified; therefore the general result is unsupported rather than shown false. For the final specialization f(r)=rn/τf(r)=r^{n/\tau}, the omitted monotonicity does hold, so the stated Hausdorff-dimension formula is unaffected.

02Proofs3 reported findingsContains incorrect or incomplete proofs

The geometric/dynamical correspondence and the proofs of the Dirichlet, thickness, and Lebesgue-measure results are correct. The divergence half of Theorem 1.4 applies the stated Mass Transference Principle outside its stated hypotheses. A separate displayed typo in that proof has a unique harmless correction.

Sections 2–4.3Correct and complete

The homogeneous-dynamics proof of the Lebesgue results is complete

Pages 8–20 · Theorems 1.1–1.3 and 1.5 · arXiv:1301.0989v4

The light-cone model preserves the relevant height and approximation inequalities, the cusp function is distance-like, and the flow has the mixing and measure-decay inputs required by the shrinking-target theorem. The convergence estimate separately follows from counting rational points and Borel–Cantelli. These steps give both directions of Theorem 1.3 and support the stated Dirichlet and badly approximable consequences.

Proof of Theorem 1.4Incomplete as written · verified scope repair

The Mass Transference Principle is applied without one of its hypotheses

Pages 20–21 · Theorem 4.7 and proof of Theorem 1.4 · arXiv:1301.0989v4

The proof establishes full Lebesgue measure for the limsup of the enlarged balls and then invokes Theorem 4.7. But that theorem is stated only for dimension functions satisfying monotonicity of r−nf(r)r^{-n}f(r), whereas Theorem 1.4 does not impose or prove this. Checking monotonicity of knf(ϕ(k))k^n f(\phi(k)) is not a substitute because it controls only the discrete radii used by the approximation function; the explicit oscillatory dimension function in the statement finding verifies the logical independence. A verified scope repair is to add the hypothesis that r−nf(r)r^{-n}f(r) is monotonic to Theorem 1.4. Proving the theorem under the weaker printed hypotheses instead would require a mass-transference result not supplied here. The power function used for the dimension corollary already meets the missing hypothesis.

Proof of Theorem 1.4Typo

The enlarged-radius expression has misplaced parentheses

Page 21 · first displayed series in the proof of Theorem 1.4 · arXiv:1301.0989v4

The left side of the first displayed series places the exponent 1/n1/n on ϕ(k)\phi(k) and has unmatched grouping, rendering the summand as (f(ϕ(k)1/n)n\bigl(f(\phi(k)^{1/n}\bigr)^n. The enlarged radius defined immediately above is f(ϕ(k))1/nf(\phi(k))^{1/n}, so the uniquely consistent expression is (f(ϕ(k))1/n)n=f(ϕ(k))\bigl(f(\phi(k))^{1/n}\bigr)^n=f(\phi(k)). This correction restores the displayed identity and does not affect the separate missing-hypothesis finding.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1301.0989v4
Authors listed
Dmitry Kleinbock, Keith Merrill
Audit date
August 19, 2026
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