Quantitative Polynomial Wiener-Wintner Theorems

TL;DR

This paper establishes quantitative polynomial Wiener-Wintner theorems for measure-preserving actions of nilpotent Lie groups, extending ergodic theory results to more general and weighted settings using advanced harmonic analysis tools.

Contribution

It introduces a broad, quantitative version of polynomial Wiener-Wintner theorems applicable to nilpotent Lie group actions, leveraging a generalized polynomial Carleson theorem.

Findings

Proved quantitative polynomial Wiener-Wintner theorems for nilpotent Lie group actions

Extended results to ergodic averages with singular integral weights

Utilized a generalized polynomial Carleson theorem in the proof

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.