Convergence of multiple ergodic averages

TL;DR

This paper discusses the convergence of multiple ergodic averages and explores their connections to classical and recent results in additive number theory, providing insights suitable for a general mathematical audience.

Contribution

It offers an accessible overview of mean convergence of multiple ergodic averages and links ergodic theory with additive number theory.

Findings

Proves mean convergence of multiple ergodic averages

Describes relationships between ergodic theory and additive number theory

Provides educational notes for a general audience

Abstract

These notes are based on a course for a general audience given at the Centro de Modeliamento Matem\'atico of the University of Chile, in December 2004. We study the mean convergence of multiple ergodic averages, that is, averages of a product of functions taken at different times. We also describe the relations between this area of ergodic theory and some classical and some recent results in additive number theory.