arXiv:2607.27582v1

Polynomial maps which are not good for nice recurrence and applications

Rigoberto Zelada

math.DS37A0505D1037A1537A4611B30

Paper metadata

Original submission: July 30, 2026
Claimed MSC: 37A05, 05D10, 37A15, 37A46, 11B30
arXiv categories: math.DS
Paper license: CC BY 4.0 ↗
Comments: 33 pages

Abstract

Let $\mathbb F_2$ be the finite field with two elements and let $\mathbb F_2^ω$ denote the countably infinite-dimensional vector space over $\mathbb F_2$ . We show that, unlike the case of polynomial maps $p:\mathbb Z\rightarrow\mathbb Z$ vanishing at zero which are always good for nice recurrence, there are polynomials $p:\mathbb F_2^ω\rightarrow \mathbb F_2^ω$ with $p(0_{\mathbb F_2^ω})=0_{\mathbb F_2^ω}$ which fail to have this property. This disproves a conjecture of Bergelson and McCutcheon (c. 2000), which predicted that for any countably infinite abelian groups $H$ and $G$ , every polynomial map $p:H\to G$ with $p(0_H)=0_G$ is good for nice recurrence. Moreover, we develop a dynamical mechanism which shows that the magnitude of intersections along polynomial paths is degree-sensitive (even when one considers only weakly mixing systems). Among other things, we also show that the Furstenberg-Sarkozy theorem for $\mathbb F_2^ω$ -valued polynomials of degree at most $d$ vanishing at zero is equivalent to a $d$ -dimensional symmetric-difference weakening of the density polynomial Hales-Jewett conjecture. Thus, as we explain in detail in this paper, our observations not only shed new light on the phenomenon of polynomial recurrence but also constrain possible strategies for proving or disproving the density polynomial Hales-Jewett conjecture.

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 18, 2026

01Statements4 reported findingsCorrect

The main construction and its three advertised consequences are supported. Theorem A produces a single weakly mixing action and exact degree- $d$ polynomial with uniformly poor two-set recurrence on suitable infinite subgroups. Corollary B disproves nice recurrence in degree $d>1$ ; Corollary C gives the stated two-sided bound on the best universal constant; Corollary D transfers the example to a positive-density set with no nontrivial popular differences along $p_d$ .

Theorem ACorrect

Uniform failure of nice polynomial recurrence in a weakly mixing action

Pages 4–5 · Theorem A · arXiv:2607.27582v1

For every $d$ , Theorem A constructs one weakly mixing $\mathbb{F}_2^\omega$ -action, sets $A_\delta$ of every measure $0<\delta\le 1/2$ , and a degree- $d$ polynomial $p_d$ . After restriction by an injective homomorphism depending on $\varepsilon$ , every nonzero polynomial value has intersection measure within $\varepsilon\delta$ of $\delta/2^d$ , while zero is the sole exceptional parameter. The construction verifies $p_d(0)=0$ and exact algebraic degree $d$ . The quantifiers are uniform in $\delta$ as printed, and the action is not changed when $\delta$ or $\varepsilon$ changes.

Corollaries B and CCorrect

Failure of nice recurrence and the sharp universal scale

Pages 5–6 · Corollaries B–C · arXiv:2607.27582v1

Taking $\delta=1/2$ and $\varepsilon=1/8$ in Theorem A gives an intersection strictly below $\mu(A)^2$ by a fixed amount for every nonzero parameter, proving Corollary B. For Corollary C, the construction yields the displayed upper bound on $a_d$ , while the polynomial Hales–Jewett lower estimate supplies the other side. The algebra in (1.4) and the resulting $O(2^{-d})$ scale agree with those substitutions, including the degree-one endpoint.

Corollary DCorrect

Dense set with no popular polynomial differences

Pages 6–7 · Corollary D · arXiv:2607.27582v1

The correspondence construction applied to the set $A_\delta$ in Theorem A produces a subset $E$ of $\mathbb{F}_2^\omega$ with the asserted upper Banach density. Its two-point correlations along a suitable Følner sequence reproduce the measure intersections. Choosing $\varepsilon$ below the uniform deficit makes every nonzero $p_d(\xi)$ fail the $\varepsilon$ -popular-difference inequality. The qualitative syndetic recurrence asserted immediately before the corollary follows separately from the universal lower bound, so the two conclusions are compatible.

Theorem A′ and finite formulationCorrect

Equivalent fixed-polynomial and finite models

Pages 23–27 · Section 6 and Appendix C · arXiv:2607.27582v1

Section 6 moves the $\varepsilon$ -dependence from the polynomial restriction into an isomorphic weakly mixing action while keeping the same degree- $d$ polynomial, giving the stated alternative formulation. The finite version is obtained by truncating the independent-coordinate model after the error parameters have been fixed. Its normalization matches the density and symmetric-difference formulation used in Appendix C; no compactness step strengthens a finite conclusion to an unjustified uniform one.

02Proofs3 reported findingsCorrect

The product system and weak-mixing verification. The recursive polynomial $q_d$ is analyzed by iterated finite differences: all derivatives of order $d+1$ vanish and one $d$ -fold derivative is nonzero, proving exact degree. On a finite Bernoulli block its values impose the correlation $\delta/2^d$ . Infinitely many independent blocks are assembled into a product action, and the injective map $\varphi_{d,\varepsilon}$ sends each finite set of nonzero parameters to fresh blocks where the error is below $\varepsilon\delta$ . The character criterion for $\mathbb{F}_2^\omega$ -actions shows that every nontrivial eigencharacter is destroyed on a later independent coordinate, hence the product action is weakly mixing. Cylinder approximation makes the estimates simultaneous for the continuum of $\delta$ -level sets constructed in the model.

Polynomial constructionCorrect and complete

The product system and weak-mixing verification

Pages 15–27 · Sections 4–6 · arXiv:2607.27582v1

The recursive polynomial $q_d$ is analyzed by iterated finite differences: all derivatives of order $d+1$ vanish and one $d$ -fold derivative is nonzero, proving exact degree. On a finite Bernoulli block its values impose the correlation $\delta/2^d$ . Infinitely many independent blocks are assembled into a product action, and the injective map $\varphi_{d,\varepsilon}$ sends each finite set of nonzero parameters to fresh blocks where the error is below $\varepsilon\delta$ . The character criterion for $\mathbb{F}_2^\omega$ -actions shows that every nontrivial eigencharacter is destroyed on a later independent coordinate, hence the product action is weakly mixing. Cylinder approximation makes the estimates simultaneous for the continuum of $\delta$ -level sets constructed in the model.

Sections 4–5Correct and complete

Correlation computation and asymptotic subgroup selection

Pages 15–23 · Sections 4–5 · arXiv:2607.27582v1

On each finite block, the event $A_\delta$ is chosen so that the zero translate has measure $\delta$ while every nonzero $q_d$ -translate intersects it with the explicitly computed $\delta/2^d$ . Independent replication makes the block errors summable. The subgroups $H_{d,\varepsilon}=\varphi_{d,\varepsilon}(\mathbb{F}_2^\omega)$ are chosen after the finite error tolerance, so all their nonzero points are governed by sufficiently late blocks. This proves the supremum estimate (1.3) and the stronger pointwise assertion in Theorem A with the quantifier order stated.

Corollaries B–DCorrect and complete

Dynamical and combinatorial deductions

Pages 5–7 and 18–23 · deductions of Corollaries B–D · arXiv:2607.27582v1

The numerical choice in Corollary B leaves a positive gap below $\mu(A)^2$ . The upper estimate for $a_d$ is the same example, and the cited recurrence theorem gives the lower estimate at the displayed density threshold. For Corollary D, the Furstenberg correspondence is used in the direction that realizes dynamical intersection bounds as upper-density correlations along a selected Følner sequence. Because the deficit is uniform over all nonzero $\xi$ , one $\varepsilon$ works for the entire polynomial image.

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:2607.27582v1
Authors listed: Rigoberto Zelada
Audit date: August 18, 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.

Audit controls

Status: Report ready
Source: arXiv v1
Prompt: Standard prompt · locked

The report is available below and in the public summary.

Review prompt

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. For every adverse or unverified outcome, restate the exact claim and hypotheses, identify its location in the reviewed version, and give the decisive reasoning. Do not infer that a theorem is false merely because its printed proof is defective. Assign the overall statements status by the following priority: 1. Contains wrong statements, if at least one central statement is Incorrect. 2. Contains unsupported statements, if no central statement is Incorrect but at least one is Not able to verify. 3. Correct, if every central statement is Correct, apart from any typographical or minor formal findings allowed by Part 4. PART 2 — CORRECTNESS AND COMPLETENESS OF THE PROOFS Audit the proofs of the central statements and every material internal or cited input on which they depend. For each proof or proof segment, use the narrowest supported outcome: - Correct and complete: every substantive step is verifiable and no case or crucial nontrivial step is missing. - Incorrect as written: a displayed calculation, inference, lemma, or cited application is formally false or inapplicable. - Incomplete as written: a necessary substantive step or case is missing, even though no explicit contradiction has been proved. - Not able to verify: the available paper and sources are insufficient to decide whether the step is valid. Do not flag omitted routine details, compressed arguments, or “analogously” steps when the omitted reasoning is recoverable and correct. 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

...

for inline mathematics and

...

for displayed mathematics, with balanced delimiters and braces. Do not leave raw LaTeX commands outside math delimiters. 12. Separate theorem truth from the correctness of the printed proof. A theorem can be Correct in Part 1 because of a fully verified independent proof while its printed proof is adverse in Part 2. Conversely, a defective proof alone never licenses an Incorrect theorem classification. Organize the report under Parts 1–3 and apply Part 4 throughout. Do not add editorial commentary, scores, rankings, recommendations, or a list of suppressed observations.

This panel is visibly marked as owner-only for the concept prototype. Authentication is not enabled locally.

Polynomial maps which are not good for nice recurrence and applications

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Uniform failure of nice polynomial recurrence in a weakly mixing action

Failure of nice recurrence and the sharp universal scale

Dense set with no popular polynomial differences

Equivalent fixed-polynomial and finite models

The product system and weak-mixing verification

Correlation computation and asymptotic subgroup selection

Dynamical and combinatorial deductions

Challenge an audit finding

Authenticate with ORCID, then require an exact arXiv match