Random Sequences and Pointwise Convergence of Multiple Ergodic Averages

Abstract : We prove pointwise convergence, as $N\to \infty$, for the multiple ergodic averages $\frac{1}{N}\sum_{n=1}^N f(T^nx)\cdot g(S^{a_n}x)$, where $T$ and $S$ are commuting measure preserving transformations, and $a_n$ is a random version of the sequence $[n^c]$ for some appropriate $c>1$. We also prove similar mean convergence results for averages of the form $\frac{1}{N}\sum_{n=1}^N f(T^{a_n}x)\cdot g(S^{a_n}x)$, as well as pointwise results when $T$ and $S$ are powers of the same transformation. The deterministic versions of these results, where one replaces $a_n$ with $[n^c]$, remain open, and we hope that our method will indicate a fruitful way to approach these problems as well.
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https://hal.archives-ouvertes.fr/hal-00543176
Contributor : Emmanuel Lesigne <>
Submitted on : Saturday, April 16, 2011 - 12:08:20 PM
Last modification on : Tuesday, July 2, 2019 - 1:37:02 AM
Long-term archiving on : Sunday, July 17, 2011 - 2:22:53 AM

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  • HAL Id : hal-00543176, version 3
  • ARXIV : 1012.1130

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Nikos Frantzikinakis, Emmanuel Lesigne, Mate Wierdl. Random Sequences and Pointwise Convergence of Multiple Ergodic Averages. Indiana University Mathematics Journal, Indiana University Mathematics Journal, 2012, 61 (2), pp.585-617. ⟨hal-00543176v3⟩

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