Random Sequences and Pointwise Convergence of Multiple Ergodic Averages

TL;DR

This paper establishes pointwise convergence for multiple ergodic averages involving random and deterministic polynomial sequences, advancing understanding of convergence behavior in ergodic theory.

Contribution

It proves new pointwise and mean convergence results for averages with random polynomial sequences, extending prior deterministic cases and suggesting new research directions.

Findings

Proved pointwise convergence for averages with random polynomial sequences.

Established mean convergence for averages with random polynomial sequences.

Provided partial results for deterministic polynomial sequences, opening new avenues for research.

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.