Random Sequences and Pointwise Convergence of Multiple Ergodic Averages

Abstract

We prove pointwise convergence, as $N\to \infty$, for the multiple ergodic averages $\frac{1}{N}{\sum }_{n=1}^{N}f(T^{n}x)\cdot g(S^{a_n}x)$, 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.

In Version 2, references have been added. In Version 3, a section on general negative results for recurrence and convergence in the case of non commuting transformations has been added