arXiv:2002.00433v3
On badly approximable numbers
math.NT11J13
Abstract
Motivated by a wonderful paper by Ngoc Ai Van Nguyen, Anthony Poëls and Damien Roy, where a powerful method was introduced, we prove a criterion for a vector to be a badly approximable vector. Moreover we construct certain examples which show that a more general version of our criterion is not valid.
AI-generated audit
Audit summary
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
01Statements2 reported findingsContains unsupported statements
Theorems 1 and 2 are supported by the detailed best-approximation construction. The separately announced dual Theorem 3 is not proved in the paper, so its correctness could not be verified from the submitted manuscript.
Theorems 1–2Correct
The simultaneous best-approximation constructions are correct
Pages 3–5 and 7–29 · Theorems 1–2 and their proofs · arXiv:2002.00433v3
The nested primitive vectors and empty cylinders give linearly independent coordinates, bounded consecutive best-denominator ratios in the required regime, and failure of the uniform badly approximable lower bound. The determinant and exclusion estimates identify the complete sequence of best approximations.
Full paper, version 3 ↗Theorem 3Not able to verify
The dual linear-form assertion is announced without a proof
Page 4 · Theorem 3 and the paragraph immediately following it · arXiv:2002.00433v3
The manuscript explicitly states that it does not give a proof of Theorem 3 and reserves the more technical dual construction for a future paper. No counterexample was found, but the text contains neither a proof nor a cited source proving the same assertion, so this theorem remains unsupported in this audit.
02Proofs2 reported findingsContains unverified proofs
The proofs supplied for Theorems 1 and 2 are complete. Theorem 3 has no proof in the manuscript and is therefore unverified rather than shown incorrect.
Proof of Theorem 1Correct and complete
Both best-approximation criteria imply bad approximability
Pages 7–12 · Sections 6–8 and Lemmas 1–2 · arXiv:2002.00433v3
For simultaneous approximation, the covolume ratios of the successive rational subspaces telescope and the bounded denominator ratios give a uniform lower bound for . For the dual criterion, the rescaled lattice converts the decreasing linear-form errors into the same geometric estimate; the final rational-subspace case is handled in its minimal dimension. These arguments prove the two nontrivial implications in Theorem 1.
Theorem 3Not supplied
No proof is supplied for the dual construction
Page 4 · announced Theorem 3 · arXiv:2002.00433v3
The authors explicitly describe Theorem 3 as an announcement. A complete audit would require the promised dual induction, including an empty-body argument showing that the constructed vectors are exactly the linear-form best approximations. Those steps are absent from this version.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.