MathAudit

Paper metadata

Original submission
November 12, 2004
Claimed MSC
arXiv categories
math.NT
Comments
27 pages

Abstract

This brief survey deals with multi-dimensional Diophantine approximations in sense of linear form and with simultaneous Diophantine approximations. We discuss the phenomenon of degenerate dimension of linear subspaces generated by the best Diophantine approximations. Originally most of these results have been established by the author in few years ago. Here we collect all of them together and give some new formulations.

AI-generated audit

Audit summary

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
01Statements3 reported findingsContains unsupported statements

Several surveyed dimension and direction theorems are supported, but the supplied arguments do not verify the all-best-approximations conclusion in Theorem 1.4 or the realization theorem for prescribed directions.

Theorem 1.4Not able to verify

The auxiliary approximations are not connected to every best approximation

Pages 4–9 · Theorem 1.4, Lemmas 1.1–1.2, and final inference · arXiv:math/0411293v1

The theorem quantifies over every best approximation and compares ζν\zeta_\nu with ψ(Mν+r−1)\psi(M_{\nu+r-1}). Lemma 1.2 constructs a separate sequence of integer linear forms with small values, but the final paragraph merely asserts that the same shifted-index inequality follows for all best approximations. It gives no index interlacing, linear-independence, or norm comparison that would control rr consecutive best approximations between two constructed forms. That is a nontrivial missing obligation, and no independent proof is supplied here.

Theorem 2.6Not able to verify

The prescribed-direction realization is only an unclosed induction sketch

Pages 21–23 · Theorem 2.6 and Section 2.12 · arXiv:math/0411293v1

The sketch chooses a rational two-plane and a nearby new integer point, then asserts that the accumulated list is exactly the full best-approximation sequence. It does not prove exclusion of every competing lattice point, preservation of all earlier approximation inequalities, convergence with the requested direction error, or rational independence of the limit. Those are the central obligations of the theorem rather than routine omitted computations.

Theorems 2.3–2.5Correct

Successive-direction separation for strictly convex norms

Pages 16–20 · Theorems 2.3–2.5 · arXiv:math/0411293v1

The empty best-approximation cylinder gives the normalized direction exclusion in Theorem 2.3. Strict convexity supplies a uniform radial loss near the touching boundary, while exponential denominator growth contradicts an indefinitely small loss. The same argument in bounded blocks, combined with bad approximability, proves Theorem 2.5.

02Proofs3 reported findingsContains incorrect or incomplete proofs

Two central construction arguments omit their decisive best-approximation controls, and Lemma 1.1 invokes an extra decay property not included in its hypotheses.

Lemma 1.1Incomplete as written · plausible repair only

The nested-interval construction assumes unprovided fast decay

Pages 5–7 · proof of Lemma 1.1 · arXiv:math/0411293v1

The lemma is stated for an arbitrary decreasing ψ(y)→0\psi(y)\to0, but nestedness is justified only after assuming Wψ(pν+1)<ψ(pν)W\psi(p_{\nu+1})<\psi(p_\nu) while also fixing pν+1≍pν/ψ(pν)p_{\nu+1}\asymp p_\nu/\psi(p_\nu). A slowly varying function need not satisfy that extra inequality. One can plausibly begin Theorem 1.4 with a faster minorant of ψ\psi, but the paper does not formulate that reduction or propagate it through the shifted best-approximation estimate.

Proofs of Theorems 1.4 and 2.6Incomplete as written · no verified repair supplied

The decisive sequence-identification steps are missing

Pages 9 and 22–23 · final proof of Theorem 1.4 and induction step for Theorem 2.6 · arXiv:math/0411293v1

For Theorem 1.4, the existence of some small linear forms is not enough to obtain the claimed inequality for every indexed best approximation. For Theorem 2.6, proximity in an absolutely rational plane does not by itself exclude all competing integer points or prove that no additional best approximations appear. Both conclusions are asserted without the needed uniform lattice exclusions.

Lemma 1.1 notationTypo

Several rr-dimensional indices are printed with ss or with the wrong coordinate

Pages 5–8 · construction and determinant calculation in Lemmas 1.1–1.2 · arXiv:math/0411293v1

Replace the occurrences of the undeclared dimension ss by rr, use the matching coordinate index in the nested intervals, and restore the missing plus sign in the first-coordinate interval. The symmetric construction and the following r×rr\times r determinant uniquely determine these corrections; they do not resolve the substantive gaps above.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:math/0411293v1
Authors listed
Nikolai G Moshchevitin
Audit date
August 20, 2026
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