Pointwise characteristic factors for Wiener Wintner double recurrence theorem

TL;DR

This paper extends Bourgain's double recurrence theorem to Wiener-Wintner averages, proving their convergence and establishing a uniform result related to the Conze-Lesigne factor in ergodic systems.

Contribution

It introduces pointwise characteristic factors for Wiener-Wintner double recurrence, generalizing previous results and providing uniform convergence criteria.

Findings

Wiener-Wintner double recurrence averages converge almost everywhere.

Uniform convergence holds when one function is orthogonal to the Conze-Lesigne factor.

The results are independent of the parameter t in the averages.

Abstract

In this paper, we extend Bourgain's double recurrence result to the Wiener-Wintner averages. Let $(X, \mathcal{F}, \mu, T)$ be a standard ergodic system. We will show that for any $f_1, f_2 \in L^\infty(X)$, the double recurrence Wiener-Wintner average \[ \frac{1}{N} \sum_{n=1}^N f_1(T^{an}x)f_2(T^{bn}x) e^{2\pi i n t} \] converges off a single null set of $X$ independent of $t$ as $N \to \infty$. Furthermore, we will show a uniform Wiener-Wintner double recurrence result: If either $f_1$ or $f_2$ belongs to the orthogonal complement of the Conze-Lesigne factor, then there exists a set of full measure such that the supremum on $t$ of the absolute value of the averages above converges to $0$.