arXiv:2303.17134v1

Measure theoretic laws for limsup set defined by rectangles

Dmitry Kleinbock, Baowei Wang

math.NT11J8311K60

Paper metadata

Original submission: March 30, 2023
Claimed MSC: 11J83, 11K60
arXiv categories: math.NT
Paper license: CC0 1.0 ↗

Abstract

In this paper, we present a general principle for the Lebesgue measure theory of limsup sets defined by rectangles under the hypothesis of ubiquity for rectangles.

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 19, 2026

01Statements2 reported findingsCorrect

The rectangle-ubiquity measure theorem and its shrinking-target and weighted Diophantine applications are correct after a fixed-constant repair to the printed covering construction. The repair changes no hypothesis, exponent, or conclusion.

Theorem 2.5Correct

The rectangle-ubiquity divergence principle is valid

Page 5 and Section 3 (pages 7--12) · measure-theory theorem and proof · arXiv:2303.17134v1

At level $n$ , ubiquity supplies on the order of $\prod_i(r_B/\rho_i(u_n))^{\delta_i}$ separated large rectangles. The scaling property supplies on the order of $\prod_i(\rho_i(u_n)/\psi_i(u_n))^{\delta_i\kappa_i}$ small rectangles inside each. Their total measure is therefore comparable to $\mu(B)\prod_i\left(\frac{\psi_i(u_n)}{\rho_i(u_n)}\right)^{\delta_i(1-\kappa_i)}.$ The pair-correlation estimate decays geometrically when either family is regular. With the fixed enlargement constants repaired as described in Part 2, the Chung--Erdos bound gives a positive proportion in every ball, and doubling gives full measure.

ApplicationsCorrect

The shrinking-target and weighted linear-form laws follow

Pages 5--6 and Sections 4--6 (pages 12--21) · Theorems 2.6--2.7 and corollary · arXiv:2303.17134v1

For digit-expansion targets the resonant rectangles tile the product space at every level, so the product series is exact. For simultaneous approximation and systems of linear forms, Minkowski's theorem supplies the rectangular ubiquity radii; the imposed geometric growth of the denominator bounds gives regularity. Substitution into the general product criterion yields the displayed convergence and divergence series.

02Proofs3 reported findingsContains incorrect or incomplete proofs

The printed proof incorrectly asserts that the fivefold enlargements selected by the covering lemma are disjoint. A verified repair replaces fixed enlargement factors and preserves every measure and correlation estimate. The Chung--Erdos denominator also omits its diagonal indices in one displayed statement, although the application includes them.

Steps 1--2 of the main proofIncorrect as written · verified repair

The covering lemma does not make the fivefold enlargements disjoint

Pages 8--9 · construction of the large and shrunk rectangle families · arXiv:2303.17134v1

The stated $5r$ lemma produces a disjoint selected subfamily $\mathcal F$ and the covering $\bigcup\mathcal G\subset\bigcup_{R\in\mathcal F}5R$ ; it does not imply that the sets $5R$ are pairwise disjoint. The proof nevertheless claims pairwise disjointness of the fivefold large rectangles and, again, of the fivefold small rectangles. Apply the same lemma to the family of fivefold rectangles: the selected $5R$ are disjoint and the original family is covered by the corresponding $25R$ . In Step 1 the $25R$ remain inside $B$ for all sufficiently large levels because every $\rho_i(u_n)$ tends to zero. In Step 2 one may replace the displayed parent rectangle by a fixed enlargement of it; since $\psi_i(u_n)\leq\rho_i(u_n)$ , the $25R$ lie in that enlargement. Ahlfors regularity and the scaling property change all resulting measures only by fixed constants. Every subsequent cardinality, intersection, series, and full-measure estimate therefore remains valid.

Chung--Erdos inequalityTypo

The displayed second-moment denominator excludes the diagonal

Page 7 · Lemma 3.1 · arXiv:2303.17134v1

The standard inequality has denominator $\sum_{1\leq i,j\leq N}\mu(E_i\cap E_j)$ , including $i=j$ . The paper prints $i\ne j$ , which is false for disjoint events. Step 4 separately retains the diagonal contribution $\sum_i\mu(E_i)$ , so changing the displayed index condition to all pairs $i,j$ is uniquely determined and makes the stated lemma agree with its actual use.

Correlation estimateCorrect and complete after verified repair

The repaired construction still gives the required quasi-independence

Pages 9--12 · Steps 3--4 · arXiv:2303.17134v1

Disjointness at the repaired fixed enlargement scale bounds the number of level- $n$ rectangles meeting a fixed level- $m$ rectangle. Splitting coordinates according to whether $\psi_i(u_m)$ exceeds $\rho_i(u_n)$ gives the two displayed factorizations. Regularity of $\rho$ or of $\Psi$ supplies a geometric factor $\lambda^{(n-m)\varepsilon}$ , whose double sum is bounded by the first moments, exactly as required by the corrected second-moment inequality.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

Detailed audit reportFull reasoning, manuscript locations, and references.

Open report PDF ↗

Author response

Challenge an audit finding

Local workflow preview

A listed author may submit formal evidence that an audit is inaccurate. The response would be considered in a fresh AI re-evaluation; it would not edit the audit automatically.

Paper: arXiv:2303.17134v1
Authors listed: Dmitry Kleinbock, Baowei Wang
Audit date: August 19, 2026

01Establish identityMatch an authenticated scholarly identity to this paper.
02Submit evidenceIdentify the finding and give a formal mathematical response.
03Re-evaluateA separate agent checks the response and records a disposition.

Email fallback for papers without a linked ORCID

A production fallback could send a one-time link only when the submitted address matches an independently maintained author-contact allowlist for this paper. MathAudit must return the same message for every address so the form cannot reveal which contacts are on that list.

Structured response preview

This form remains unavailable until production identity verification succeeds. Nothing entered here is submitted.

Audit finding challengedWhy the evaluation is inaccurateFormal evidence or repairVerifiable sources, if anyI am submitting evidence, not instructions for the reviewing agent.

This panel never establishes authorship in the local prototype. A production result should be described narrowly as an authenticated ORCID match or control of a separately allowlisted author-contact mailbox.

Measure theoretic laws for limsup set defined by rectangles

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The rectangle-ubiquity divergence principle is valid

The shrinking-target and weighted linear-form laws follow

The covering lemma does not make the fivefold enlargements disjoint

The displayed second-moment denominator excludes the diagonal

The repaired construction still gives the required quasi-independence

Challenge an audit finding

Authenticate with ORCID, then require an exact arXiv match