Powers of sequences and convergence of ergodic averages

Nikos Frantzikinakis; Michael Johnson; Emmanuel Lesigne; Mate Wierdl

hal-00327604, version 4

Powers of sequences and convergence of ergodic averages

Nikos Frantzikinakis ¹, Michael Johnson () ², Emmanuel Lesigne () ³, Mate Wierdl () ¹

Ergodic Theory and Dynamical Systems 30, 5 (2010) 1431-1456

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.

1: Department of Mathematics and Statistics
University of Victoria
2: Department of Mathematics and Statistics, Swarthmore College
Swarthmore College
3: Laboratoire de Mathématiques et Physique Théorique (LMPT)
CNRS : UMR6083 – Université François Rabelais - Tours

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From: Emmanuel Lesigne <>
Submitted on: Monday, 29 June 2009 16:36:43
Updated on: Saturday, 9 February 2013 09:00:56

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arXiv: 0810.1581

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