A note on uniform convergence of Wiener-Wintner ergodic averages

TL;DR

This paper proves uniform convergence of Wiener-Wintner ergodic averages for ergodic actions of locally compact abelian groups and introduces a finitary van der Corput inequality for these groups.

Contribution

It establishes uniform convergence results for Wiener-Wintner averages in a broad class of group actions and provides a new finitary inequality for analysis.

Findings

Proved uniform convergence of Wiener-Wintner averages for LCA group actions.

Derived a finitary van der Corput inequality for locally compact abelian groups.

Extended ergodic theory results to more general group settings.

Abstract

We show uniform convergence of Wiener-Wintner ergodic averages for ergodic actions of (not necessarily countable) locally compact, second countable, abelian (LCA) groups. As a by-product, we obtain a finitary version of the van der Corput inequality for such groups.