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Paper metadata

Original submission
April 28, 2025
Latest revision
October 6, 2026 · version 3
Claimed MSC
37B05, 37B20, 37A45
arXiv categories
math.DS
Paper license
CC BY 4.0 ↗
Comments
Some changes have been made correcting minor issues. In particular, the statement of Theorem C has changed slightly

Abstract

In this work we give a full characterization of sets of multiple polynomial recurrence in Weyl systems, which are ergodic unipotent affine transformations on products of tori and finite abelian groups. In particular, we show that measurable and topological recurrence in Weyl systems coincide. Our analysis also yields a structure theorem for polynomial multicorrelation sequences in Weyl systems. These results stem from an in-depth study of the Weyl complexity of a set of polynomials and the introduction of a new concept: the \textit{Weyl polynomials} generated by a set of polynomials.

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Audit summary

Audited against arXiv v3

Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues 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 October 24, 2026
01Statements2 reported findingsNo statement issues found

The current Weyl-system recurrence characterization, stabilization threshold, polynomial-frequency structure, and uncorrelatedness criterion are supported. Extensions to arbitrary systems are retained as conjectures or explicitly conditional consequences.

Theorems A–C; Propositions 3.9, 3.17No issue found

Weyl polynomial spaces control the claimed recurrence and correlation structure

PDF pp. 13–23, 31–35 · arXiv:2504.19899v3

The annihilator condition on the polynomial matrix identifies precisely the frequencies surviving integration of characters. Its rational linear algebra gives integral bases and stabilization at the Weyl complexity. Rectangular Fourier truncations approximate each bounded function in the required finite LpL^p norm, uniformly controlling the resulting correlations; surviving character products have the stated polynomial phases and square-summable coefficient sequence. This is enough for the liminf-of-uniform-tail approximation asserted in the structure theorem.

Corollary D; Theorem 4.8; Corollaries 4.10–4.12No issue found

Both directions of the recurrence and independence equivalences check

PDF pp. 25–27, 32–36 · arXiv:2504.19899v3

Disjoint rational polynomial spaces prevent cancellation of irrational phase coefficients, while a common nonzero integral polynomial produces correlated character examples. Topological recurrence near zero forces all finitely many Weyl frequencies near zero. Conversely the IP recurrence lower bound ensures positivity near the zero polynomial phase, and Kronecker recurrence chooses arbitrarily large indices with those phase constraints. A polynomial outside one Weyl space gives the explicit separating set used for optimality and the inclusion criterion.

02Proofs2 reported findingsNo proof issues found

The Kronecker-preserving extension, Fourier estimates, and reduction through finite connected components close the principal proof chain in v3.

Lemmas 2.6–2.7; Proposition 3.13No issue found in the reviewed proof

The algebraic extension and polynomial-basis changes preserve the required data

PDF pp. 8–10, 16–17 · arXiv:2504.19899v3

Connectedness of the kernel makes the Smith-normal-form diagonal entries of the Kronecker factor map equal to one, supplying an integral right inverse. The columns C,NC,…,Nd−1CC,NC,\ldots,N^{d-1}C span the ambient space and intertwine the unipotent maps; the affine translation adjustment then gives a surjective factor with the same Kronecker rotation. Vandermonde's identity supplies the displayed invertible change from binomial to divided-power polynomial matrices.

Theorems 3.18 and 4.8No issue found in the reviewed proof

Residue restrictions and the connected-component reduction do not lose recurrence

PDF pp. 23–25, 33–34 · arXiv:2504.19899v3

A sufficiently divisible progression makes all shifted, divided polynomials integral. The invariance of Weyl polynomial membership under that change is established before its use. Partition regularity selects a residue class retaining the full Kronecker recurrence property, and a finite rotation forces every relevant exponent to preserve the connected component. The matrix I+A+⋯+Al−1I+A+\cdots+A^{l-1} has nonzero determinant lDl^D, so its torus map is onto and produces an ergodic affine root of the return map. The connected proof consequently applies with all required divisibility and measure factors intact.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper
arXiv:2504.19899v3
Authors listed
Felipe Hernández
Audit date
October 24, 2026
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