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A sequence $(s_n)$ of integers is good for the mean ergodic theorem if foreach invertible measure preserving system $(X,\mathcal{B},\mu,T)$ and anybounded measurable function $f$, the averages $ \frac1N \sum_{n=1}^Nf(T^{s_n}x)$ converge in the $L^2$ norm. We construct a sequence $(s_n)$ thatis 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 alsoprove a similar result for pointwise convergence of single ergodic averages.

Powers of sequences and convergence of ergodic averages