Higher-Dimensional Moving Averages and Submanifold Genericity

TL;DR

This paper extends the theory of multi-parameter ergodic averages to higher dimensions, providing conditions for convergence and exploring the genericity of submanifold averages in \,\mathbb R^d.

Contribution

It generalizes previous results to \,\mathbb R^d, characterizes convergence conditions, and examines the non-convergence of averages along certain submanifolds.

Findings

Necessary and sufficient conditions for pointwise convergence of averages over boxes.

Averages along dilates of locally flat submanifolds may not converge.

Connection to submanifold-genericity concept.

Abstract

We generalize results of Jones and Olsen on multi-parameter moving ergodic averages to measure-preserving actions of $\mathbb R^d$ for $d\geq 1$. In particular, we give necessary and sufficient conditions for the pointwise convergence of averages over families of boxes in $\mathbb R^d$. As an application of our characterization, we show that averages along dilates of "locally flat" submanifolds in $\mathbb R^d$ do not necessarily converge point-wise for bounded measurable functions. This is closely related to the concept of submanifold-genericity recently introduced in \cite{BFK25}.