# 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 $(X,\mathcal{B},\mu,T)$ 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.
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Journal articles

Cited literature [14 references]

https://hal.archives-ouvertes.fr/hal-00327604
Contributor : Emmanuel Lesigne <>
Submitted on : Monday, June 29, 2009 - 4:36:43 PM
Last modification on : Thursday, August 22, 2019 - 1:13:17 AM
Long-term archiving on : Friday, September 24, 2010 - 11:20:00 AM

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### Identifiers

• HAL Id : hal-00327604, version 4
• ARXIV : 0810.1581

### Citation

Nikos Frantzikinakis, Michael Johnson, Emmanuel Lesigne, Mate Wierdl. Powers of sequences and convergence of ergodic averages. Ergodic Theory and Dynamical Systems, Cambridge University Press (CUP), 2010, 30 (5), pp.1431-1456. ⟨hal-00327604v4⟩

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