(Uniform) Convergence of Twisted Ergodic Averages

TL;DR

This paper extends the Wiener-Wintner theorem to uniform convergence of twisted ergodic averages involving Hardy field functions and polynomial phases, providing new convergence results for these averages in ergodic theory.

Contribution

It introduces uniform convergence results for twisted ergodic averages with Hardy field weights and polynomial phases, expanding classical ergodic theorems.

Findings

Proved uniform Wiener-Wintner type theorems for Hardy field weights.

Established pointwise convergence of twisted polynomial averages for all  in [0,1].

Provided elementary proof techniques for convergence in ergodic averages.

Abstract

Let $T$ be an ergodic measure-preserving transformation on a non-atomic probability space $(X,\Sigma,\mu)$. We prove uniform extensions of the Wiener-Wintner theorem in two settings: For averages involving weights coming from Hardy field functions, $p$: \[ \{\frac{1}{N} \sum_{n\leq N} e(p(n)) T^{n}f(x) \} \] and for "twisted" polynomial ergodic averages: \[ \{\frac{1}{N} \sum_{n\leq N} e(n \theta) T^{P(n)}f(x) \} \] for certain classes of badly approximable $\theta \in [0,1]$. We also give an elementary proof that the above twisted polynomial averages converge pointwise $\mu$-a.e. for $f \in L^p(X), \ p >1,$ and arbitrary $\theta \in [0,1]$.