arXiv:2607.18987v2

A refined structure theorem for polynomial return-time sets in minimal systems

Ioannis Kousek

math.DS37B2037A0537B05

Paper metadata

Original submission: July 21, 2026
Latest revision: August 13, 2026 · version 2
Claimed MSC: 37B20, 37A05, 37B05
arXiv categories: math.DS
Paper license: CC BY 4.0 ↗
Comments: 22 pages

Abstract

We prove a refinement of a recent structural result in topological dynamics due to Glasscock, Koutsogiannis, Le, Moreira, Richter, and Robertson, who showed that in a minimal system, polynomial return-times differ by non-piecewise syndetic sets from those in its maximal infinite-step pronilfactor. In particular, we prove that the infinite-step pronilfactor can be replaced by the maximal $k$ -step pronilfactor of the system, where $k$ depends only on the given polynomials, with $k-1$ being equal to the Host-Kra complexity of the polynomials.

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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

01Statements2 reported findingsCorrect

The finite-step refinement for polynomial return-time sets in Theorem A and the simultaneous recurrence statement in Theorem B are correct. The factor step is one larger than the Host-Kra characteristic complexity for the normalized polynomial family, as stated.

Theorem ACorrect

Finite-step pronilfactors control the return-time difference

Pages 2 and 8-18 · Theorem A, Theorem 3.1, and Appendix A · arXiv:2607.18987v2

For the maximal infinite-step pronilfactor, Theorem 3.2 gives simultaneous recurrence for every finite family of normalized shifts $p_i(n+a)-p_i(a)$ . Lemma 2.2 then converts syndeticity of all finite intersections of shifts of the original return-time set into non-piecewise-syndeticity of the difference. Appendix A combines the finite-step saturation relation with the earlier infinite-step structure theorem through the open and almost-one-to-one factor diagrams, yielding the required interiors for a general minimal system. The integer used is the pronilfactor one step above the uniformity seminorm order controlling the polynomial averages.

Glasscock et al., infinite-step structure theorem ↗

Theorem B / Theorem 3.2Correct

Simultaneous recurrence over a common finite-step factor

Pages 3 and 9-13 · Theorem B and Theorem 3.2 · arXiv:2607.18987v2

The Host-Kra seminorm estimate replaces each open-set indicator by its conditional expectation on $Z_k$ with vanishing Cesaro error. Relative independent joining over $Z_k$ has support equal to the full fibre relation, so the assumed common fibre point yields a positive-measure set on which all conditional expectations are bounded below. Polynomial multiple recurrence on the product system then gives a positive average of the product of all $m$ recurrence terms, and hence a common return time with positive upper Banach density.

Full paper, version 2 ↗

02Proofs5 reported findingsCorrect

The characteristic-factor reduction, relative-joining support argument, simultaneous polynomial recurrence, combinatorial syndeticity step, and the transfer from the infinite-step factor to arbitrary minimal systems are correct and complete. Two mechanically determined notation errors are reported in yellow and do not affect any conclusion.

Theorems 3.3 and 3.2Correct and complete

Uniformity control and product recurrence

Pages 9-13 · Section 3.2 · arXiv:2607.18987v2

The tensor-square argument lowers the uniformity order by one, ergodic decomposition permits application of Leibman's characteristic-factor theorem on almost every component, and dominated convergence restores the original system. Telescoping then replaces every indicator by its $Z_k$ -conditional expectation. Proposition 3.6 identifies the support of the relative joining with the fibre relation, and the Bergelson-Leibman theorem applied to $A_1\times\cdots\times A_m$ supplies the strict positive lower bound required in (17).

Full paper, version 2 ↗

Theorem 3.1Correct and complete

Syndetic finite intersections imply the claimed small difference

Pages 13-15 · Section 3.3 · arXiv:2607.18987v2

For a finite $F=\{a_1,\ldots,a_m\}$ in the factor return set, normalized shifts of the polynomial tuple satisfy Theorem 3.2 simultaneously. Minimality aligns the resulting nonempty open intersections. Adding distinct linear terms produces one essentially distinct tuple, and the IP-polynomial recurrence theorem makes its return set syndetic. The explicit containment of its translate in $\bigcap_{a\in F}(B-a)$ verifies the hypothesis of Lemma 2.2 and proves (10).

Full paper, version 2 ↗

Definition of thick setsTypo

The finite interval should be contained in $A$

Page 4 · first bullet list in Section 2.1 · arXiv:2607.18987v2

The printed definition says that for every $N$ there is $m$ with $\{m,m+1,\ldots,m+N\}\subset\mathbb Z$ , a condition independent of $A$ . The uniquely intended correction is $\{m,m+1,\ldots,m+N\}\subset A$ . All subsequent uses, including Lemmas 2.1-2.2, employ the standard corrected definition, so no argument is affected.

Full paper, version 2 ↗

Definition of $q_{j,i}$Typo

The normalized shift includes $a_j$ twice in the first expression

Pages 13-14 · proof of Theorem 3.1, definition immediately after (18) · arXiv:2607.18987v2

Since $p_{j,i}(n)=p_i(n+a_j)-p_i(a_j)$ , the displayed identity is correct only with $q_{j,i}(n)=p_{j,i}(n+n_0)-p_{j,i}(n_0)=p_i(n+n_0+a_j)-p_i(n_0+a_j).$ The printed first expression instead uses $n+n_0+a_j$ and $n_0+a_j$ as arguments of $p_{j,i}$ , inserting $a_j$ a second time. Removing those two extra occurrences is mechanically forced by the following equality and by every downstream use. The corrected $q_{j,i}$ still vanishes at $0$ and has the required degree and distinctness properties.

Full paper, version 2 ↗

Appendix ACorrect and complete

Transfer through the open-extension diagram

Pages 16-18 · Appendix A · arXiv:2607.18987v2

The continuity-point argument upgrades Theorem 3.2 to the fibre-saturation inclusion (21). The corresponding inclusion for the infinite-step factor follows from the cited structure theorem through Veech's open extension, and the almost-one-to-one fibres preserve a dense $G_\delta$ set. Intersecting the two residual sets and using continuity of the fibre maps gives (24). Applying (24) inside the nonempty interiors of the projected open sets supplies the common return time needed for the reduction in A.1.

Glasscock et al., version 2 ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2607.18987v2
Authors listed: Ioannis Kousek
Audit date: August 18, 2026

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A refined structure theorem for polynomial return-time sets in minimal systems

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Finite-step pronilfactors control the return-time difference

Simultaneous recurrence over a common finite-step factor

Uniformity control and product recurrence

Syndetic finite intersections imply the claimed small difference

The finite interval should be contained in AAA

The normalized shift includes aja_jaj​ twice in the first expression

Transfer through the open-extension diagram

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Authenticate with ORCID, then require an exact arXiv match

The finite interval should be contained in $A$

The normalized shift includes $a_j$ twice in the first expression