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arXiv:1310.6691v1

Sur une question de N. Chevallier liée à l'approximation Diophantienne simultanée

Nikolay Moshchevitin

math.NT11J13
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Paper metadata

Original submission
October 24, 2013
Claimed MSC
11J13
arXiv categories
math.NT
Comments
10 pages, in French

Abstract

We prove a conjecture due to Nicolas Chevallier concerning unimodular matrices related to simultaneous Diophantine approximation to real numbers.

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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.

Current report

Detailed mathematical audit

Generated August 20, 2026
01Statements2 reported findingsCorrect

The construction answering Chevallier's question and the companion existence theorem for unimodular approximation triples are supported by the paper's nested-cone and best-approximation arguments.

Theorem 1 and Proposition 2Correct

The obstruction to uniformly good unimodular triples is verified

Sections 1 and 3-5 · Theorem 1, Proposition 2, and the inductive construction · arXiv:1310.6691v1

The construction chooses nested cones and complete rank-two lattices so that every sufficiently accurate integer vector belongs to the current plane. Lemma 5 then forces any third vector completing a unimodular matrix to have error bounded below by the prescribed scale. Choosing the scales against the given decreasing function yields Theorem 1, while the same determinant obstruction gives Proposition 2.

Theorem 2Correct

Unimodular triples with errors tending to zero exist for every independent pair

Sections 1 and 6 · Theorem 2 · arXiv:1310.6691v1

Each consecutive pair of best approximations spans a complete rank-two lattice and can be extended to a basis of Z3\mathbb Z^3. Projecting its fundamental parallelogram to the neighboring affine lattice plane gives an integer point completing the pair to a unimodular triple. The projection offset equals the reciprocal covolume and tends to zero by the cited covolume-error relation, so all three approximation errors tend to zero.

02Proofs3 reported findingsCorrect

The geometric induction and determinant estimates close the stated arguments. A few coordinate symbols are evident typos and do not change any conclusion.

Lemmas 1-4Correct and complete

The cone and complete-lattice induction preserves all invariants

Sections 2-4 · Lemmas 1-4 and conditions (i)-(viii) · arXiv:1310.6691v1

The local cone lemmas provide an open neighborhood in which the designated lattice vectors remain consecutive best approximations. Lemma 4 supplies the next primitive vector and complete plane, while the shrinking diameters give a limiting vector with the required rational independence.

Lemma 5Correct after typos

The determinant lower bound is valid after two mechanical symbol repairs

Section 5 · Lemma 5 and its displayed determinant · arXiv:1310.6691v1

Expanding the nonzero integral determinant in the two approximation-error coordinates gives the claimed lower bound once the third entry in the first row is read as a2−qξ2a_2-q\xi_2 rather than a2−qξ1a_2-q\xi_1, and the last entry is read as a2,ν−qνξ2a_{2,\nu}-q_\nu\xi_2. The parallel rows uniquely determine both corrections.

Lemma 5 displayTypos

Two mismatched coordinates are typos

Section 5 · determinant in the proof of Lemma 5 · arXiv:1310.6691v1

The repeated ξ1\xi_1 in the third error coordinate must be ξ2\xi_2, and qν2q_{\nu2} must be qνq_\nu. Both are isolated notation mismatches; the subsequent norm estimate uses the corrected two-coordinate vector.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1310.6691v1
Authors listed
Nikolay Moshchevitin
Audit date
August 20, 2026
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