Powers of sequences and convergence of ergodic averages

Abstract

A sequence $(s_n)$ of integers is good for the mean ergodic theorem if for each invertible measure preserving system $(X,\mathcal{B},\hat{I}¼,T)$ and any bounded measurable function $f$, the averages $\frac{1}{N}{\sum }_{n=1}^{N}f(T^{s_n}x)$ 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.

After a few minor corrections, to appear in Ergodic Theory and Dynamical Systems