arXiv:1510.06214v3

Eine Bemerkung über positiv definite quadratische Formen und rationale Punkte

Nikolay Moshchevitin

math.NT11J1311E04

Paper metadata

Original submission: October 21, 2015
Latest revision: January 9, 2016 · version 3
Claimed MSC: 11J13, 11E04
arXiv categories: math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: The paper is written in German. In the second version we corrected the constant. Minor corrections in the third version

Abstract

We give an elementary proof of a recent result by Fishman, Kleinbock, Merrill and Simmons about rational points on quadratic surfaces.

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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

01Statements3 reported findingsCorrect

The effective rational-approximation conclusions for isotropic quadratic forms are mathematically supported for sufficiently large $T$ . The printed range $T\geq1$ has a small-denominator boundary exception and needs a simple global range correction.

Satz 1Correct · minor formal correction

The effective quadratic-surface approximation estimate is verified in its operative range

Sections 1 and 3-4 · Satz 1 and its proof · arXiv:1510.06214v3

Minkowski's successive-minima bound and Cassels' small-zero theorem produce a nonzero integral isotropic vector in $C_f\mathfrak K_f(\alpha,t)$ . The hyperbolic automorphism then gives $q\ll C_ft$ and $f(\alpha-a/q)\ll C_f/(qt)$ ; taking $t=2T/(3C_f)$ yields the displayed constant $\kappa_f=6C_f^2$ whenever $t$ is in the large-parameter range used by the inequalities.

Range $T\geq1$Minor formal correction

The lower endpoint is false for forms with no denominator-one rational point

Sections 1 and 4 · hypotheses of Satz 1 and Satz 2 · arXiv:1510.06214v3

Let $f(x_1,x_2)=2x_1^2+2x_2^2$ . Then $F=f-y^2$ is isotropic because $F(1,1,2)=0$ , and $f(1/\sqrt2,0)=1$ . For $T=1$ , however, the conclusion requires $q=1$ and $2a_1^2+2a_2^2=1$ , which is impossible. Replacing $T\geq1$ by $T\geq T_0(f)$ , as the proof itself requires, repairs both main statements without affecting their asymptotic content.

Satz 2Correct · same range correction

The independent-points extension follows from the cited small-zero theorem

Section 5 · Satz 2 and Schulze-Pillot theorem · arXiv:1510.06214v3

Applying the stated Schulze-Pillot bound to the induced integral quadratic form supplies $n+1$ linearly independent isotropic coefficient vectors with effective heights. Their images remain independent and lie in one effective dilation of the same approximation body, so the calculation for Satz 1 applies to each. The only necessary change is the same lower cutoff on $T$ .

02Proofs3 reported findingsCorrect

The geometry-of-numbers and isotropic-zero arguments are complete after restricting the small external parameter range; repeated display labels and coordinate slips are typographical only.

Successive-minima dichotomyCorrect and complete

Both cases produce an effective integral isotropic point

Section 3 · equations (16)-(22) · arXiv:1510.06214v3

If the first minimum is below one, integrality and $|F(g)|<1$ force $F(g)=0$ . Otherwise Minkowski bounds the last minimum; pulling back the isotropic form to a lattice basis and applying Cassels bounds a nonzero isotropic combination. The coefficient and dilation estimates combine to the stated $C_f$ .

Section 4 parameter substitutionBoundary issue · repaired by the theorem range

The proof implicitly requires a lower bound on $t$

Section 4 · equations (25)-(29) and $t=2T/(3C_f)$ · arXiv:1510.06214v3

The step $2C_f/t+2C_ft<3C_ft$ requires $t$ bounded away from zero. Thus it proves the conclusion only once $T$ exceeds a form-dependent constant. This matches the explicit denominator obstruction above and is repaired by changing the displayed range everywhere it is used.

Section 4 notationTypos · no status impact

Three local display errors have unique corrections

Section 4 · definition of $t$ , the bound for $q+v_n$ , and the expansion of $F_0$ · arXiv:1510.06214v3

The definition of $t=2T/(3C_f)$ is missing a closing denominator brace; the bound printed as $|q+v_n|<{2C_f}{t}$ must be $|q+v_n|<2C_ft$ ; and one expansion starts with the nonexistent coordinate $v_0$ instead of $v_1$ . The immediately following calculations use the corrected expressions.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:1510.06214v3
Authors listed: Nikolay Moshchevitin
Audit date: August 20, 2026

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Eine Bemerkung über positiv definite quadratische Formen und rationale Punkte

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The effective quadratic-surface approximation estimate is verified in its operative range

The lower endpoint is false for forms with no denominator-one rational point

The independent-points extension follows from the cited small-zero theorem

Both cases produce an effective integral isotropic point

The proof implicitly requires a lower bound on ttt

Three local display errors have unique corrections

Challenge an audit finding

Authenticate with ORCID, then require an exact arXiv match

The proof implicitly requires a lower bound on $t$