Powers of sequences and convergence of ergodic averages

TL;DR

This paper constructs specific integer sequences demonstrating that being good for the mean ergodic theorem can depend on the sequence's powers, revealing nuanced behaviors in ergodic averages.

Contribution

It introduces sequences with tailored properties for ergodic convergence, including sequences where powers are selectively good or bad for the mean ergodic theorem.

Findings

Constructed sequences where $(s_n)$ is good but $(s_n^2)$ is not.

For any set of bad exponents, found sequences that are good exactly outside that set.

Extended results to multiple and pointwise ergodic averages.

Abstract

A sequence $(s_n)$ of integers is good for the mean ergodic theorem if for each invertible measure preserving system $(X,\mathcal{B},\mu,T)$ and any bounded measurable function $f$, the averages $ \frac1N \sum_{n=1}^N f(T^{s_n}x)$ converge in the $L^2$ norm. We construct a sequence $(s_n)$ that is good for the mean ergodic theorem, but the sequence $(s_n^2)$ is not. Furthermore, we show that for any set of bad exponents $B$, there is a sequence $(s_n)$ where $(s_n^k)$ is good for the mean ergodic theorem exactly when $k$ is not in $B$. We then extend this result to multiple ergodic averages. We also prove a similar result for pointwise convergence of single ergodic averages.