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Quantitative Polynomial Wiener-Wintner Theorems

Lars Becker, Asgar Jamneshan, Christoph Thiele

TL;DR

The paper advances Wiener–Wintner theory by proving quantitative, r-variation boundedness for variably modulated ergodic averages on doubling metric measure spaces and, in particular, on homogeneous nilpotent Lie groups. It combines a generalized polynomial Carleson theorem with a Calderón transference framework to handle polynomial phases (via Leibman polynomials) and singular-integral weights, achieving uniform finite r-variation bounds for both averaged and singular-weighted averages. The approach extends the classical result to a broad geometric setting and yields sparse bounds that extend the p-range to (1,∞), with concrete specialization to homogeneous Lie groups and implications for polynomial maps into unitary groups. This provides a robust, quantitative refinement of Wiener–Wintner-type convergence with potential applications to ergodic theory on noncommutative groups and to harmonic-analytic modulation phenomena in geometric settings.

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.

Quantitative Polynomial Wiener-Wintner Theorems

TL;DR

The paper advances Wiener–Wintner theory by proving quantitative, r-variation boundedness for variably modulated ergodic averages on doubling metric measure spaces and, in particular, on homogeneous nilpotent Lie groups. It combines a generalized polynomial Carleson theorem with a Calderón transference framework to handle polynomial phases (via Leibman polynomials) and singular-integral weights, achieving uniform finite r-variation bounds for both averaged and singular-weighted averages. The approach extends the classical result to a broad geometric setting and yields sparse bounds that extend the p-range to (1,∞), with concrete specialization to homogeneous Lie groups and implications for polynomial maps into unitary groups. This provides a robust, quantitative refinement of Wiener–Wintner-type convergence with potential applications to ergodic theory on noncommutative groups and to harmonic-analytic modulation phenomena in geometric settings.

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.
Paper Structure (29 sections, 22 theorems, 145 equations)

This paper contains 29 sections, 22 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Wiener--Wintner theorems
  3. Main result on metric measure spaces
  4. Metric measure space
  5. Modulation functions
  6. Cancellation
  7. Singular integral kernels
  8. Statement of the main result
  9. Homogeneous Lie groups
  10. Remarks
  11. Variation estimates
  12. Generality of the acting group
  13. Lattices
  14. Structure of the paper
  15. Proof of the variational Wiener--Wintner theorem with smooth cutoffs
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $(X,\nu,T)$ be a measure-preserving system, and let $f \in L^1(X,\nu)$. Then there exists a full measure set $X_f \subset X$ such that for every $\theta \in [0,2\pi]$ and $x \in X_f$, the averages converge as $N \to \infty$, where $e(t) \coloneqq e^{it}$.

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Proposition 2.1
  • Theorem 2.2: beckeretal, Theorem 1.1
  • Proposition 2.3
  • ...and 24 more