Convergence of multiple ergodic averages along cubes for several commuting transformations

TL;DR

This paper proves the convergence of multiple ergodic averages along cubes for several commuting transformations, using the magic extension method, and derives related combinatorial results.

Contribution

It establishes the norm convergence of multiple ergodic averages along cubes for multiple commuting transformations using a novel approach.

Findings

Proves norm convergence of multiple ergodic averages along cubes.

Derives combinatorial results from ergodic theory.

Utilizes the magic extension method by B. Host.

Abstract

We prove the norm convergence of multiple ergodic averages along cubes for several commuting transformations, and derive corresponding combinatorial results. The method we use relies primarily on the "magic extension" established recently by B. ~Host.