A Hard-Analytic Proof of "Most" Polynomial Wiener-Wintner Theorems for Infinite Measure Spaces

TL;DR

This paper introduces a new hard-analytic proof for most cases of the polynomial Wiener-Wintner theorem in infinite measure spaces, establishing convergence of certain polynomial-modulated ergodic averages.

Contribution

It provides the first hard-analytic proof for the polynomial Wiener-Wintner theorem in $\sigma$-finite measure spaces, covering linear and degree-two polynomials.

Findings

Convergence of polynomial-modulated averages for linear polynomials.

Convergence for quadratic polynomials vanishing at zero.

Applicable to $\sigma$-finite measure-preserving systems.

Abstract

We provide a new proof of ``most" cases of the polynomial Wiener-Wintner theorem for $\sigma$-finite spaces, using hard-analytic methods. Specifically, we prove that whenever $(X,\mu,T)$ is a $\sigma$-finite measure-preserving system, and $f \in L^p(X), \ 1 \leq p < \infty$, there exists a co-null set $X_f \subset X$ so that for all $\omega \in X_f$ \[ \frac{1}{N} \sum_{n \leq N} e^{2 \pi i P(n)} f(T^n \omega) \] converges for all polynomials $P$ which are either linear, or vanish to degree $2$ at the origin.