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arXiv:2509.25628v3

A note on general isolation result in Diophantine Approximation

Sergei Pitcyn, Nikolay Moshchevitin

math.NT11J0411J0611J13
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Paper metadata

Original submission
September 30, 2025
Latest revision
October 15, 2025 · version 3
Claimed MSC
11J04, 11J06, 11J13
arXiv categories
math.NT

Abstract

In the present paper we give very simple general statements which deal with approximation of a real number by rationals and are related to isolation phenomenon. In particular we study functions f(x)>f1(x)>0 f(x)>f_1(x)>0 such that existence of solutions pq\frac{p}{q} of Diophantine inequality ∣α−pq∣<f(q)q2 \left| α-\frac{p}{q}\right|< \frac{f(q)}{q^2} leads to the existence of solutions of inequality ∣α−pq∣<f1(q)q2 \left| α-\frac{p}{q}\right|< \frac{f_1(q)}{q^2} .

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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.

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Detailed mathematical audit

Generated August 20, 2026
01Statements2 reported findingsCorrect

The isolation principle for improvements of infinitely-often Diophantine inequalities is correct, including the matrix generalization. The proof needs a local compactness repair described under Proofs.

Theorems 1 and 3Correct

A universal positive improvement can be chosen independently of the matrix

Pages 2–3 and 7–8 · Theorems 1 and 3 · arXiv:2509.25628v3

Matrices already satisfying the half-sized inequality are handled by the fixed improvement f/2f/2. On the complementary tail sets, the family-F\mathcal F exclusion of equality gives a positive finite-level gap; taking the smaller of the two improvements proves the implication for every matrix.

Full paper, version 3 ↗
Theorem 2Correct

The rational-valued scalar case has the announced effective improvement

Pages 3 and 8–10 · Theorem 2 and Section 6 · arXiv:2509.25628v3

The recursive continued-fraction construction tests finitely many rational candidates at each denominator and chooses a positive rational gap below all active errors. The irrationality of α\alpha prevents equality with a rational threshold, so the resulting computable function is positive and gives the implication stated in Theorem 2.

02Proofs2 reported findingsContains incorrect or incomplete proofs

The category argument and continued-fraction specialization are correct. In Theorem 3, however, the displayed minimum is taken over the wrong complement and can be zero or negative; replacing it by the compact tail complement with the active inequality restores the proof.

Equation (19)Incorrect as written; verified repair

The claimed positive minimum on a single-vector complement is false

Pages 7–8 · equation (19) and proof of Theorem 3 · arXiv:2509.25628v3

The set MqcM_{\mathbf q}^{c} only excludes the half-sized inequality for one vector. It still contains matrices with ∣q∣m/n∥Θq∥=f(∣q∣)|\mathbf q|^{m/n}\lVert\Theta\mathbf q\rVert=f(|\mathbf q|), where the displayed gap is zero, and matrices with larger values, where it is negative. For x=∣q∣x=|\mathbf q|, instead minimize over the compact tail complement Mxc=⋂∣r∣≥xMrc\mathcal M_x^c=\bigcap_{|\mathbf r|\geq x}M_{\mathbf r}^c restricted to pairs satisfying ∣q∣m/n∥Θq∥≤f(x)|\mathbf q|^{m/n}\lVert\Theta\mathbf q\rVert\leq f(x). The definition of F\mathcal F excludes equality uniformly for large xx, so compactness gives a positive gap; every Θ∈Mc\Theta\in\mathcal M^c belongs to these tail complements for all sufficiently large xx. This supplies exactly the missing universal g2(x)g_2(x).

Sections 4 and 6Correct and complete

The Baire-category and one-dimensional specialization arguments are sound

Pages 4–7 and 8–10 · Lemmas 1–2 and proof of Theorem 2 · arXiv:2509.25628v3

The exceptional parameter set is a countable union of nowhere-dense images, and the rational-value hypothesis rules out the forbidden equality by linear independence. The continued-fraction estimates then give the effective scalar refinement stated in the introduction.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:2509.25628v3
Authors listed
Sergei Pitcyn, Nikolay Moshchevitin
Audit date
August 20, 2026
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