article

Title: Random sequences and pointwise convergence of multiple ergodic averages

Authors: Nikos Frantzikinakis, Emmanuel Lesigne and Mate Wierdl

Issue: Volume 61 (2012), Issue 2, 585-617

Abstract:

$We prove pointwise convergence, as $N\to\infty$, for the multiple ergodic averages $(1/N)\sum_{n=1}^N f(T^nx)\cdot g(S^{a_n}x)$, 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 $(1/N)\sum_{n=1}^N f(T^{a_n}x)\cdot g(S^{a_n}x)$, as well as pointwise results when $T$ and $S$ are powers of the same transformations. 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.$