arXiv:2601.03999v3

Quantitative Polynomial Wiener-Wintner Theorems

Lars Becker, Asgar Jamneshan, Christoph Thiele

math.DSmath.CA37A3037A4642B20

Paper metadata

Original submission: January 7, 2026
Latest revision: July 28, 2026 · version 3
Claimed MSC: 37A30, 37A46, 42B20
arXiv categories: math.DS, math.CA
Paper license: CC BY 4.0 ↗
Comments: v2: 29 pages, incorporated referee comments; v3: final version accepted for publication in Israel J. Math., incorporated minor revisions following second referee report

Abstract

We prove quantitative polynomial Wiener--Wintner theorems in a very general setup, including measure-preserving actions of nilpotent Lie groups. Our results apply both to ergodic averages and to averages with singular integral weights. The proof relies on the generalized polynomial Carleson theorem developed in the companion paper by van Doorn, Srivastava, and the authors.

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Audit summary

Audited against arXiv v3

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 four principal estimate families are supported. Theorems 1.2–1.3 establish quantitative variation/jump control for polynomial Wiener–Wintner averages and their ergodic transfer. Theorem 1.5 gives the homogeneous-Lie-group singular integral estimate. Theorem 1.6 supplies sparse domination and its weighted consequences. The $p$ , $r$ , Hölder, degree, and dilation hypotheses match the inputs used in the corresponding proofs.

Theorems 1.2 and 1.3Correct

Variational polynomial Wiener–Wintner bounds

Pages 2–5 · Theorems 1.2–1.3 · arXiv:2601.03999v3

Theorems 1.2 and 1.3 give quantitative jump/variation estimates for the polynomially modulated singular averages associated with the stated Hölder kernels. The former is formulated on the underlying metric-measure model and the latter in the transferred ergodic setting. Their $p$ , $r$ , and Hölder ranges coincide with the interpolation and companion Carleson estimates used in the proof. Uniformity is over the polynomial coefficients and truncation sequence in the precise classes displayed, while dependence on degree and structural constants is retained.

Theorem 1.5Correct

Homogeneous-group polynomial singular integrals

Pages 6–7 · Theorem 1.5 · arXiv:2601.03999v3

For a connected simply connected homogeneous nilpotent Lie group, the theorem bounds the stated polynomially parameterized singular integral with a Hölder Calderón–Zygmund kernel. The group dilations, Haar dimension, cancellation convention, and truncation are the same as in Proposition 6.1. Lifting the polynomial map to the universal nilpotent model and pushing the estimate back preserves the operator norm, so the conclusion is uniform in coefficients as stated.

Theorem 1.6 and Corollary 1.7Correct

Sparse domination and weighted consequences

Pages 7–8 · Theorem 1.6 and Corollary 1.7 · arXiv:2601.03999v3

Theorem 1.6 dominates the bilinear form of the maximal polynomial operator by a sparse form with the printed exponents. The stopping cubes are selected using the already proved local maximal estimates, and the recursion terminates on finite-support inputs. Standard sparse-form bounds then yield Corollary 1.7 with the displayed weight characteristic and do not enlarge the $p$ -range. The constants retain the paper's declared dependence on dimension, degree, and kernel regularity.

Corollary 1.4Correct

Ergodic polynomial Wiener–Wintner convergence

Page 5 · Corollary 1.4 · arXiv:2601.03999v3

The finite-family variational bound transfers to the measure-preserving action and implies that the truncation sequence is Cauchy almost everywhere. A dense class gives convergence directly, and the maximal inequality extends it to all $L^p$ inputs. The polynomial phase and coefficient quantifiers remain uniform during this approximation, yielding exactly the ergodic convergence statement.

02Proofs3 reported findingsCorrect

Frequency decomposition, sparse domination, and transference. The proof separates the dyadic truncations into major and minor frequency regimes. Weyl-type cancellation makes the minor-frequency increments summable in variation, while the major pieces are approximated by continuous companion multipliers whose errors also sum. The continuous variational Carleson input is invoked only in its allowed $p$ and $r$ range. Interpolation then gives the announced jump estimate, and a finite-truncation Calderón transference carries it to the ergodic model before any limiting assertion is used. This establishes both maximal control and convergence rather than assuming convergence to define the variation.

Variational and sparse estimatesCorrect and complete

Frequency decomposition, sparse domination, and transference

Pages 11–38 · Sections 2–6 and Appendix A · arXiv:2601.03999v3

The proof separates the dyadic truncations into major and minor frequency regimes. Weyl-type cancellation makes the minor-frequency increments summable in variation, while the major pieces are approximated by continuous companion multipliers whose errors also sum. The continuous variational Carleson input is invoked only in its allowed $p$ and $r$ range. Interpolation then gives the announced jump estimate, and a finite-truncation Calderón transference carries it to the ergodic model before any limiting assertion is used. This establishes both maximal control and convergence rather than assuming convergence to define the variation.

Sections 4–5Correct and complete

Vector-valued interpolation and sparse recursion

Pages 22–31 · Sections 4–5 · arXiv:2601.03999v3

The local vector-valued estimate controls a stopping cube by its parent averages and a summable tail. Good and bad children are disjoint and occupy at most the required fraction, so iteration produces a genuinely sparse family. Interpolation is applied to finite sequences of truncations, where the Banach-valued norms are measurable and the endpoint estimates are available. The resulting exponents are then inserted into the recursion without crossing an endpoint, proving Theorem 1.6.

Section 6Correct and complete

Nilpotent lifting and transference

Pages 31–38 · Section 6 · arXiv:2601.03999v3

Leibman's lifting represents a bounded-degree polynomial sequence by a fixed universal nilpotent translation followed by a homomorphism. Haar coordinates and homogeneous dilations transport the kernel cancellation and size bounds to that model. Proposition 6.1 is uniform in the homomorphism, so quotienting recovers the original group estimate. Compact support and finite truncation justify Fubini at the transference stage; exhaustion and Fatou then give the full operator bound.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2601.03999v3
Authors listed: Lars Becker, Asgar Jamneshan, Christoph Thiele
Audit date: August 18, 2026

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Quantitative Polynomial Wiener-Wintner Theorems

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Variational polynomial Wiener–Wintner bounds

Homogeneous-group polynomial singular integrals

Sparse domination and weighted consequences

Ergodic polynomial Wiener–Wintner convergence

Frequency decomposition, sparse domination, and transference

Vector-valued interpolation and sparse recursion

Nilpotent lifting and transference

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