arXiv:2606.17880v2

Pair correlation statistics for dynamical systems

Simon Baker, Mike Todd

math.DS37B2037A0537E0511K36

Paper metadata

Original submission: June 16, 2026
Latest revision: August 6, 2026 · version 2
Claimed MSC: 37B20, 37A05, 37E05, 11K36
arXiv categories: math.DS
Paper license: arXiv non-exclusive distribution license ↗
Comments: Some expository updates

Abstract

We study the pair correlation statistics of orbits generated by maps on the interval. We show that under suitable mixing and multifractal assumptions, the pair correlation statistics of an orbit will almost surely exhibit the same asymptotic behaviour as a suitable sequence of i.i.d. random variables. We will also show that, under suitable hypotheses, the pair correlation statistics defined by two orbits will almost surely exhibit the same behaviour as two suitable sequences of i.i.d. random variables. Specific dynamical systems to which our results apply to include Gibbs-Markov maps and the Gauss map. We also give an example of a slowly mixing system for which the pair correlation statistics of an orbit almost surely behave distinctly to an i.i.d. sequence.

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Audit summary

Audited against arXiv v2

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 findingsContains wrong statements

The conditional two-orbit and one-orbit limit theorems are correct under their printed mixing, dimension, early-return, and scaling hypotheses. The two advertised Gibbs–Markov corollaries are false for the class defined in the paper, because Definition 1.4 does not impose irreducibility or mixing. A piecewise linear map with two invariant mixing components satisfies that definition but gives component-dependent pair-correlation limits, contradicting both corollaries.

Theorem 2.1 and Theorem ACorrect

Two-orbit pair-correlation limit under the stated quantitative hypotheses

Pages 7 and 10–18 · Theorem A and Theorem 2.1 · arXiv:2606.17880v2

The second-moment decomposition separates two large time gaps, two short gaps, and the two half-separated regimes. Exponential $BV$ – $L^\infty$ mixing controls the separated terms, Lemma 2.3 and hypotheses (2.1)–(2.2) give a relative variance $O(n^{-\eta})$ for some $\eta>0$ , and the subsequence argument plus continuous mean scaling yields the all- $n$ , all- $s>0$ conclusion. The dimension inequalities in Theorem A permit parameters $\gamma<2-\beta C_{\mu_1,\mu_2}$ and $\alpha<2\gamma$ exactly when $\beta(2C_{\mu_1,\mu_2}-C_{\mu_i})<2$ for $i=1,2$ ; hence the printed range $0<\beta<2/C_{\max}$ is sufficient. The local notation corrections reported in Part 2 do not alter this argument.

Full paper, version 2 ↗

Theorem BCorrect

Single-orbit pair-correlation limit under mixing and recurrence hypotheses

Page 9 and Sections 3–4 · Theorem B · arXiv:2606.17880v2

The dyadic conditioning in Section 3 replaces moving balls by interval indicators at a smaller scale. The first moment is asymptotic to $n^2\int\mu(B(x,s/n^\beta))\,d\mu(x)$ , while the four-index decomposition in Section 4 controls the well-separated, zero-gap, one-gap, and two-gap configurations. The assumptions $\beta C_\mu<2$ , $\beta(C_\mu-D_\mu)<1$ , and $\beta(C_\mu-F_\mu)<1$ make every exceptional contribution lower order. The resulting summable-subsequence estimate and continuous mean scaling prove the stated almost-sure limit. The exponent-label typo reported in Part 2 has a unique mechanical correction and does not affect the theorem.

Full paper, version 2 ↗

Corollaries 1.5 and 1.6Incorrect

The defined Gibbs–Markov class includes nonmixing maps for which both limits fail

Pages 7–9 · Definition 1.4 and Corollaries 1.5–1.6; pages 40–41 · their proofs · arXiv:2606.17880v2

Let $I=[0,1]$ and use the four partition intervals $P_1=[0,1/4]$ , $P_2=[1/4,1/2]$ , $P_3=[1/2,3/4]$ , and $P_4=[3/4,1]$ . Define $T(x)=\begin{cases}1/2-2x,&x\in P_1,\\2x-1/2,&x\in P_2,\\2x-1/2,&x\in P_3,\\5/2-2x,&x\in P_4.\end{cases}$ The formulas agree at common endpoints. Every branch is $C^1$ , has $|T'|=2$ , zero distortion, and image diameter $1/2$ ; each image is a union of partition elements. Thus $T$ satisfies Definition 1.4 with attractor $I$ . Lebesgue measure is invariant with density $\rho=1$ , but the two halves $[0,1/2]$ and $[1/2,1]$ are invariant mixing components. Conditional on either half, the invariant density is $2$ , so the single-orbit numerator divided by $n^{2-\beta}$ tends to $4s$ , not the $2s$ asserted by Corollary 1.6. For two Lebesgue-typical points, the corresponding normalized limit is $4s$ when the points lie in the same component and $0$ when they lie in opposite components, rather than the constant $2s$ asserted by Corollary 1.5. Repair classification: add a mixing or primitive Markov hypothesis and specify the associated mixing acip; under that restriction the corollaries follow from Theorems A and B as claimed.

02Proofs4 reported findingsContains incorrect or incomplete proofs

The proofs of the conditional limit theorems are complete after three local notation corrections. The proofs of the Gibbs–Markov corollaries, however, invoke mixing and recurrence properties that do not follow from the paper's own Definition 1.4; the explicit two-component map in Part 1 shows that this is a substantive and conclusion-changing hypothesis failure.

Proofs of Corollaries 1.5 and 1.6Incorrect as written · verified repair

Mixing is asserted for a class that does not require it

Pages 40–41 · proofs of Corollaries 1.5 and 1.6 · arXiv:2606.17880v2

The proof says that every map satisfying Definition 1.4 has exponential $BV$ – $L^\infty$ mixing, and then invokes the four-mixing and early-return inputs. Definition 1.4 contains expansion, Markov, distortion, and big-image conditions but no irreducibility, aperiodicity, or mixing condition. The two-component counterexample in Part 1 satisfies every displayed defining condition and preserves Lebesgue measure, yet it is not mixing and violates both conclusions. The cited Gibbs–Markov results apply in their mixing setting and cannot supply a missing hypothesis in the paper's broader definition. The repair is verified after adding exponential $BV$ – $L^\infty$ mixing to Corollary 1.5 and, for Corollary 1.6, exponential four-mixing and the stated early-return estimate; Theorems A and B plus Lemma 5.1 then give the printed limits.

Holland–Nicol–Török, Gibbs–Markov recurrence estimate ↗

Lemma 2.2(b)Minor formal correction

The strict endpoint inequality should be non-strict

Page 11 · Lemma 2.2(b) and its proof · arXiv:2606.17880v2

The lemma concludes $\beta<(2-\gamma)/\underline C_{\mu_1,\mu_2}$ from (2.1), but its proof yields only $\beta\leq(2-\gamma)/\underline C_{\mu_1,\mu_2}$ . Equality can occur: for $\mu_1=\mu_2$ equal to Lebesgue measure, $\underline C_{\mu_1,\mu_2}=1$ and $n^2\int\mu_1(B(x,s/n^\beta))\,d\mu_2(x)\asymp n^{2-\beta}$ , so (2.1) holds with $\gamma=2-\beta$ . Replace $<$ by $\leq$ in part (b). This endpoint statement is not used in deriving Theorem A, which uses parts (a) and (c), so no main result changes.

Proof of Theorem 2.1Typo

The all-integer lower interpolation accidentally excludes equal time indices

Pages 13–14 · lower-bound interpolation following Equation (2.5) · arXiv:2606.17880v2

The two-orbit count in Theorem 2.1 and Equation (2.5) ranges over all pairs $i,j$ , but the lower interpolation displays suddenly write $i\ne j$ , and the concluding display repeats that restriction. Replace each $0\leq i\ne j\leq m^K$ there by $0\leq i,j\leq m^K$ , and make the same correction in the concluding display. Monotonicity of the unrestricted count gives the displayed lower bound directly, so the intended correction is unique and no estimate changes.

Proof of Theorem BTypo

The final variance estimate cites the wrong exponent label

Page 38 · Equations (4.9)–(4.10) and the final choice of $\delta$ · arXiv:2606.17880v2

Proposition 4.1 supplies the well-separated error exponent $\epsilon_2$ , while Proposition 4.7 supplies $\epsilon_8$ . Equation (4.9) and the final line instead use $\epsilon_3$ in place of $\epsilon_2$ . Replace the exponent in (4.9) by $\min\{\epsilon_2,\epsilon_8,1\}$ and take $\delta$ using $\min\{\epsilon_1,\epsilon_2,\epsilon_8,1\}$ . These are exactly the exponents furnished by the immediately preceding estimates, so the correction is mechanical and preserves a positive decay rate.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2606.17880v2
Authors listed: Simon Baker, Mike Todd
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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For every defect, identify its smallest exact location, explain its downstream dependency, and classify any proposed repair as one of: - Verified repair: supply enough reasoning to verify that the repair works throughout every affected use. - Plausible repair only: describe the possible repair but do not treat it as established. - No repair supplied. Assign the overall proofs status by the following priority: 1. Contains incorrect or incomplete proofs, if at least one proof is Incorrect as written or Incomplete as written. 2. Contains unverified proofs, if there is no proved incorrectness or incompleteness but at least one material proof is Not able to verify. 3. Correct, if all audited proofs are Correct and complete, apart from any typographical or minor formal findings allowed by Part 4. PART 3 — NOVELTY OF THE MAIN STATEMENTS This is a one-sided, evidence-only check. Evaluate only central results. Report a result only when a verifiable primary source, publicly available before the earliest public version of the paper that contains the result, proves the same statement up to a change of notation, or when the result follows directly and immediately from such an earlier theorem. For each non-novelty finding, provide the earlier source, its public date, the exact theorem or proposition, and an explicit hypothesis-and-conclusion mapping. A later source cannot establish non-novelty. Do not flag a result merely because an easier proof exists or because it follows through a nontrivial argument. Do not flag a result that the paper itself explicitly presents as known, a survey item, or an alternative proof. Mention only statements that meet this non-novelty threshold. Say nothing about statements for which no such evidence was found: do not certify novelty, list unsuccessful searches, or add statement-level “no prior source found” entries. 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. Do not report a gap, unsupported statement, minor formal correction, or recommendation to add an explanatory sentence, and do not mention the suppressed item in the report. This suppression does not apply to a genuine typographical error as defined below, because a typo is a literal notation defect that still requires a textual correction. For example, if

P(delta)

is proved for every sufficiently small

delta>0

and

P

is immediately monotone in the required direction, then the claim that

P(delta)

holds for every

delta>0

requires no correction; the omitted monotonicity step is exposition rather than an audit finding. 6. Immediate-consequence closure permits reconstruction of an omitted inference; it does not permit silent alteration of the printed domain, hypotheses, or quantifiers of an intermediate statement later invoked outside its stated scope. If formal validity requires changing the text of that statement or its stated range, report the appropriate finding even when the repair is easy. Thus a result stated only for

t>0

and subsequently applied at

t=0

remains a formal scope mismatch unless the endpoint case was separately established before the application. 7. Apply this mandatory gate before recording any adverse, unverified, typographical, or minor formal finding: (a) identify the exact assertion or inference alleged to be defective; (b) reconstruct the strongest conclusion following from the preceding argument after all admissible immediate formal consequences are added; (c) identify the specific defect that remains — a formally demonstrated falsehood, a concrete nontrivial unresolved proof obligation, or a genuine domain, hypothesis, quantifier, or notation mismatch requiring a textual correction; and (d) state the evidence and minimal repair. If no specific defect remains, or if the only possible addition would be explanatory, suppress the finding entirely. Use Not able to verify only when a precise, nontrivial unresolved obligation can be identified. 8. A typographical error is an incorrect symbol, index, exponent, sign, variable name, or notation mismatch whose unique intended correction is mechanically determined from the surrounding text and does not alter the mathematical argument. Report it in yellow as Typo, with the printed text, correction, exact location, and reason it is harmless. It does not lower an overall status. 9. A minor formal correction is a boundary, endpoint, or displayed parameter-range defect that can be repaired consistently in every affected statement by a local change, without a new argument, without invalidating any related result or application, and without creating another substantive defect. Report it in yellow as Minor formal correction, with the same details. It does not lower an overall status. 10. Do not use either minor category if more than one materially different repair is possible; if the repair changes the substance or scope of a central result; if an excluded case is used later; if the correction fails to propagate consistently; or if a new mathematical idea is required. In such cases use the strongest substantive classification actually supported by evidence. 11. Write every mathematical expression as valid, compilable LaTeX. Use

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Pair correlation statistics for dynamical systems

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Two-orbit pair-correlation limit under the stated quantitative hypotheses

Single-orbit pair-correlation limit under mixing and recurrence hypotheses

The defined Gibbs–Markov class includes nonmixing maps for which both limits fail

Mixing is asserted for a class that does not require it

The strict endpoint inequality should be non-strict

The all-integer lower interpolation accidentally excludes equal time indices

The final variance estimate cites the wrong exponent label

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