Pointwise convergence of averages along cubes

TL;DR

This paper proves the pointwise convergence of certain multiple averages along cubes in measure-preserving systems, extending understanding of ergodic averages involving multiple functions.

Contribution

It establishes the pointwise convergence of averages involving three and more functions along cubes, advancing ergodic theory and multiple recurrence results.

Findings

Proved pointwise convergence of three-function averages along cubes

Extended convergence results to averages with seven functions

Contributed to the theory of multiple ergodic averages

Abstract

Let $(X,\mathcal{B},\mu, T)$ be a measure preserving system. We prove the pointwise convergence of the averages $$\frac{1}{N^2}\sum_{n,m= 0}^{N-1} f_1(T^nx)f_2(T^mx)f_3(T^{n+m}x)$$ and of similar averages with seven bounded functions.