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arXiv:1808.07070v1

Rational approximation on quadrics: a simplex lemma and its consequences

Dmitry Kleinbock, Nicolas de Saxcé

math.NTmath.DS11J1311J8337A17
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Paper metadata

Original submission
August 21, 2018
Claimed MSC
11J13, 11J83, 37A17
arXiv categories
math.NT, math.DS
Comments
15 pages

Abstract

We give elementary proof of stronger versions of several recent results on intrinsic Diophantine approximation on rational quadric hypersurfaces X⊂Pn(R)X\subset \mathbb{P}^n(\mathbb{R}). The main tool is a refinement of the simplex lemma, which essentially says that rational points on XX which are sufficiently close to each other must lie on a totally isotropic rational subspace of XX.

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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.

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Detailed mathematical audit

Generated August 19, 2026
01Statements2 reported findingsCorrect

The quadric simplex lemma and the resulting extremality, Hausdorff-dimension, and winning theorems are correct. Two symbols in the simplex calculation have unique typographical repairs.

Lemma 3.1Correct

Low-height rational points lie in one totally isotropic rational subspace

Pages 6--7 · simplex lemma for quadric hypersurfaces · arXiv:1808.07070v1

The Dani estimate sends every rational point of height at most cρ−1c\rho^{-1} in a ρ\rho-ball to an integer isotropic vector of uniformly small norm. For two such vectors, integrality and the quadratic bound give ∣Q(vi±vj)∣<1|Q(\mathbf v_i\pm\mathbf v_j)|<1, hence both values vanish. Pairwise vanishing of the associated bilinear form makes their rational span totally isotropic.

Theorems 4.2, 4.6, and 4.10Correct

The three metric consequences have the stated exponents

Pages 8--12 · applications to intrinsic approximation · arXiv:1808.07070v1

Isotropic absolute decay and the simplex lemma give a summable cover of points with exponent above one. Hausdorff--Cantelli applied with the regularity and decay exponents yields the displayed dimension upper bound. In the isotropic-subspace game, deleting the subspace supplied by the simplex lemma prevents every dangerous rational approximation at the relevant scale, proving the winning result and its thickness consequence.

02Proofs3 reported findingsCorrect

The dynamical estimate, integrality argument, covering sums, and winning strategy are correct. The simplex proof contains a missing absolute-value pair and one coordinate-name typo, both mechanically repairable.

Proof of Lemma 3.1Typo

Absolute values are missing from the quadratic-form estimate

Page 7 · final display in the proof of the simplex lemma · arXiv:1808.07070v1

The preceding sentence says it is enough to prove ∣Q(vi±vj)∣<1|Q(\mathbf v_i\pm\mathbf v_j)|<1, and the chosen constant satisfies ∣Q(w)∣≤C1∥w∥2|Q(\mathbf w)|\leq C_1\|\mathbf w\|^2. The next display drops the absolute-value bars around both occurrences of QQ. Restoring them is uniquely determined, and then the displayed bound 4/5<14/5<1 proves the required integrality conclusion.

Proof of Lemma 3.2Typo

The terminal coordinate is named inconsistently

Page 6 · Dani-correspondence estimate · arXiv:1808.07070v1

The decomposition uses coordinates v1,…,vn+1v_1,\ldots,v_{n+1} and derives a bound for vn+1v_{n+1} from Q(v)=0Q(\mathbf v)=0, but the following line prints ∣vd∣|v_d|. Replacing vdv_d by vn+1v_{n+1} is forced by the decomposition and gives the third term in the asserted maximum.

Section 4Correct and complete

The metric applications use the simplex scale in the correct direction

Pages 8--12 · proofs of the application theorems · arXiv:1808.07070v1

For a height block 2k≤H(v)<2k+12^k\leq H(v)<2^{k+1}, balls have radius 2−k(1+ε/2)2^{-k(1+\varepsilon/2)}, small enough for the simplex bound and large enough to contain every approximation ball. Decay contributes the summable factor 2−kαε/22^{-k\alpha\varepsilon/2}. The dimension and game arguments use the same scale with the correct rank bound on isotropic subspaces.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1808.07070v1
Authors listed
Dmitry Kleinbock, Nicolas de Saxcé
Audit date
August 19, 2026
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