A pointwise ergodic theorem along return times of rapidly mixing systems

TL;DR

This paper introduces a new class of sparse sequences generated by return times in rapidly mixing systems, proving their ergodic and universal convergence properties for ergodic averages, extending classical results like Bourgain's Return Times Theorem.

Contribution

It establishes that certain return time sequences in rapidly mixing systems are ergodic and universally $L^2$-good, generalizing previous results to non-independent sequences.

Findings

Sequences generated by return times are ergodic and pointwise universally $L^2$-good.

A specific class of sequences involving powers of 2 and fractional parts are ergodic for almost every point.

The approach adapts techniques from independent random variables to dependent, rapidly mixing systems.

Abstract

We introduce a new class of sparse sequences that are ergodic and pointwise universally $L^2$-good for ergodic averages. That is, sequences along which the ergodic averages converge almost surely to the projection to invariant functions. These sequences are generated randomly as return or hitting times in systems exhibiting a rapid correlation decay. This can be seen as a natural variant of Bourgain's Return Times Theorem. As an example, we obtain that for any $a\in (0,1/2)$, the sequence $\left\{n\in\mathbb{N}:\ 2^ny\mod{1}\in (0,n^{-a})\right\}$ is ergodic and pointwise universally $L^2$-good for Lebesgue almost every $y\in [0,1]$. Our approach builds on techniques developed by Frantzikinakis, Lesigne, and Wierdl in their study of sequences generated by independent random variables, which we adapt to the non-independent case.