Convergence of Diagonal Ergodic Averages

TL;DR

This paper provides an ergodic theory proof of Tao's convergence theorem for multiple ergodic averages involving commuting transformations, using novel variations of the Furstenberg correspondence to translate combinatorial recurrence results.

Contribution

It introduces two new variations of the Furstenberg correspondence that facilitate ergodic proofs of convergence theorems for multiple averages.

Findings

Established ergodic proof of Tao's convergence theorem

Developed two new Furstenberg correspondence variations

Connected combinatorial recurrence with ergodic properties

Abstract

Tao has recently proved that if $T_1,...,T_l$ are commuting, invertible, measure-preserving transformations on a dynamical system then for any $L^\infty$ functions $f_1,...,f_l$, the average $\frac{1}{N}\sum_{n=0}^{N-1}\prod_{i\leq l}f_i\circ T^n_i$ converges in the $L^2$ norm. Tao's proof is unusual in that it translates the problem into a more complicated statement about the combinatorics of finite spaces by using the Furstenberg correspondence "backwards". In this paper, we give an ergodic proof of this theorem, essentially a translation of Tao's argument to the ergodic setting. In order to do this, we develop two new variations on the usual Furstenberg correspondence, both of which take recurrence-type statements in one dynamical system and give equivalent statements in a different dynamical system with desirable properties.