arXiv:2607.13659v1

Area correlations related to lattice points in discs

Matteo Bordignon, Pär Kurlberg

math.NT11H0611P2111K0637A25

Paper metadata

Original submission: July 15, 2026
Claimed MSC: 11H06, 11P21, 11K06, 37A25
arXiv categories: math.NT
Paper license: CC BY 4.0 ↗
Comments: 30 pages, 1 figure

Abstract

Motivated by the Lester-Wigman vanishing area correlation conjecture for lattice points near the boundary of circles of growing radius, we investigate the dynamics of circle flows on tori (which are related to the motion of a charged particle in a magnetic field on a torus.) We show that an analogue of the vanishing correlation conjecture holds in this setting, i.e., we have "mixing for the area observable" despite the flow being essentially integrable. We also determine the probability density function of the areas, in global as well as local regimes.

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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 18, 2026

01Statements3 reported findingsCorrect

The correlation theorem and global distribution laws are supported. Theorem 1 proves asymptotic decorrelation in its stated joint range. Proposition 2 computes the limiting mean and variance of the area variable. Proposition 3 identifies the global limiting density from the continuum parallel-line model. A missing no-vertex-crossing hypothesis in Lemma 4 is a local formal correction and does not affect the circular-arc applications.

Theorem 1Correct

Vanishing local correlations for the lattice-circle area process

Pages 1–3 · Theorem 1 · arXiv:2607.13659v1

In the simultaneous range of radius, separation $k$ , and interval length specified in Theorem 1, the centered areas cut out by successive lattice-circle intersections have correlation tending to zero. The quantitative error is uniform in the admissible interval and $k$ ranges, with $Q=\min\{\log R,\sqrt{k}/\log k\}$ . The continuum comparison uses the same normalization as the discrete area variables. The standing exclusion of lattice points on the circle removes simultaneous horizontal/vertical crossings, including the minor Lemma 4 issue recorded below.

Propositions 2–3Correct

Global moments and limiting area density

Pages 3–4 and 24–29 · Propositions 2–3 · arXiv:2607.13659v1

Averaging the local continuum law over slope and intercept gives the density printed in Proposition 3. Its integral is one, and direct integration gives the mean and second moment used in Proposition 2. The discrete-to-continuum error is integrable uniformly over the circle, while the excluded low-denominator arcs have vanishing weight. Thus the moment limits and density use the same normalization as Theorem 1.

Lemma 4Minor formal correction · no status impact

The monotone-curve square count needs a no-vertex-crossing hypothesis

Page 6 · Lemma 4, second assertion · arXiv:2607.13659v1

As written for an arbitrary coordinatewise monotone curve, the assertion is false: the segment from $(0,0)$ to $(n,n)$ meets only the diagonal half-open squares, not $2n+O(1)$ squares. The intended applications are circular arcs, and the standing assumption $C(R)\cap\mathbb{Z}^2=\varnothing$ excludes simultaneous horizontal and vertical grid crossings. Adding that no-vertex-crossing hypothesis repairs every use without changing a main result.

02Proofs3 reported findingsCorrect

Equidistribution of boundary intersections and decorrelation. The circle is decomposed into Farey arcs according to the slope denominator. Low-height exceptional slopes have negligible total length. On every remaining short arc, rational-grid equidistribution replaces the discrete sequence of cell crossings by the continuum model, with an error uniform in the allowed separation $k$ . In that model the relevant parallel-line area variable has mean one half and the separated variables factor asymptotically. Summing the arc estimates proves Theorem 1; integrating the one-point formulas gives the global mean, variance, and density in Propositions 2–3.

Farey and continuum reductionCorrect and complete

Equidistribution of boundary intersections and decorrelation

Pages 9–29 · Sections 3–7 · arXiv:2607.13659v1

The circle is decomposed into Farey arcs according to the slope denominator. Low-height exceptional slopes have negligible total length. On every remaining short arc, rational-grid equidistribution replaces the discrete sequence of cell crossings by the continuum model, with an error uniform in the allowed separation $k$ . In that model the relevant parallel-line area variable has mean one half and the separated variables factor asymptotically. Summing the arc estimates proves Theorem 1; integrating the one-point formulas gives the global mean, variance, and density in Propositions 2–3.

Sections 5–7Correct and complete

Farey dissection, equidistribution, and global averaging

Pages 16–29 · Sections 5–7 · arXiv:2607.13659v1

Farey neighbours control how the circle crosses consecutive grid lines and isolate arcs with small rational slope denominator. The exceptional arcs satisfy the required total-length bound. On the complement, exponential-sum/equidistribution estimates are uniform in $Q$ and $k$ , and the choice $Q=\min\{\log R,\sqrt{k}/\log k\}$ makes every error term vanish. The continuum integral then gives decorrelation and, after one-point averaging, the global law.

Proof of Lemma 4Minor formal correction · no status impact

Simultaneous grid crossings are not counted

Page 6 · proof of Lemma 4 · arXiv:2607.13659v1

The proof adds horizontal and vertical crossings as if they were always distinct. They are distinct for the circle arcs used later because the circle contains no lattice point. Stating that restriction in the lemma makes the count and all downstream applications valid.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2607.13659v1
Authors listed: Matteo Bordignon, Pär Kurlberg
Audit date: August 18, 2026

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You are a mathematical auditor. Evaluate only the correctness of the paper's central statements, the correctness and completeness of its proofs, and evidence-backed non-novelty. Do not make editorial comments or evaluate importance, exposition, style, or suitability for publication. Keep the report neutral, precise, and professional. PART 1 — CORRECTNESS OF THE MAIN STATEMENTS Identify the paper's central statements and assess each one separately using exactly one of these outcomes: - Correct: the statement has a correct, verifiable proof; or, despite a defect in the printed proof, you can give a fully verified repair or independent proof. - Incorrect: you can give strong formal evidence that the statement as written is false, preferably a counterexample or a complete contradiction. - Not able to verify: the supplied proof does not establish the statement, and you can neither supply a fully verified proof nor disprove it. 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If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. An immediate formal consequence is a local, mechanically checkable inference using only definitions already in force, elementary logic, direct substitution or renaming, weakening of an inequality or set inclusion, or a monotonicity or symmetry property explicit in a definition or already proved. 4. Such an inference must introduce no new lemma, estimate, construction, case analysis, limiting or compactness argument, external result, or new issue of uniformity or parameter dependence. Check the direction of monotonicity and the dependence of every threshold or constant explicitly. Do not exchange pointwise and uniform conclusions, change exceptional sets, interchange quantifiers or limits, or extend an endpoint under this rule unless the required fact was already established. 5. If the claimed conclusion follows after immediate-consequence closure, it is fully supported. 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P(delta)

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P(delta)

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Area correlations related to lattice points in discs

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Vanishing local correlations for the lattice-circle area process

Global moments and limiting area density

The monotone-curve square count needs a no-vertex-crossing hypothesis

Equidistribution of boundary intersections and decorrelation

Farey dissection, equidistribution, and global averaging

Simultaneous grid crossings are not counted

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