On the almost everywhere convergence of two-parameter ergodic averages along directional rectangles

TL;DR

This paper investigates the almost everywhere convergence of two-parameter ergodic averages over rectangles with specific directional properties, establishing convergence for lacunary slopes and divergence for non-lacunary directions.

Contribution

It provides new results on convergence and divergence of ergodic averages along rectangles with particular directional slopes, expanding understanding of multi-parameter ergodic theory.

Findings

Convergence of averages with lacunary slopes in all L^p spaces.

Construction of divergence examples for non-lacunary directions.

Identification of directional conditions affecting convergence.

Abstract

In this paper, we study the almost everywhere convergence of sequences of two-parameter ergodic averages over rectangles in the plane. On the one hand, we show that if the rectangles we consider have their sides with slopes in a finitely lacunary set, then the averages converge almost everywhere in all $L^p$ spaces, $1 < p \leq \infty$. On the other hand, given some non-lacunary sets of directions, we construct sequences of rectangles oriented along these directions for which the associated ergodic averages fail to converge almost everywhere in any $L^p$ space, $1 < p < \infty$.