Multivariable averaging on sparse sets

TL;DR

This paper extends nonstandard ergodic averages to higher dimensions and more complex groups, establishing pointwise convergence results for sparse group averages on Z^d, including deterministic and random cases.

Contribution

It generalizes one-dimensional nonstandard ergodic theorems to Z^d and virtually nilpotent groups, introducing new variants of sparse averages with convergence guarantees.

Findings

Established L^1 pointwise ergodic theorems for sparse averages on Z^d

Extended results for sparse block and random averages to virtually nilpotent groups

Introduced native and product averages with convergence on Z^d

Abstract

Nonstandard ergodic averages can be defined for a measure-preserving action of a group on a probability space, as a natural extension of classical (nonstandard) ergodic averages. We extend the one-dimensional theory, obtaining L^1 pointwise ergodic theorems for several kinds of nonstandard sparse group averages, with a special focus on the group Z^d. Namely, we extend results for sparse block averages and sparse random averages to their analogues on virtually nilpotent groups, and extend Christ's result for sparse deterministic sequences to its analogue on Z^d. The second and third results have two nontrivial variants on Z^d: a "native" d-dimensional average and a "product" average from the 1-dimensional averages.