Multivariable averaging on sparse sets
Dynamical Systems
2012-06-21 v1 Probability
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.
Cite
@article{arxiv.1206.4381,
title = {Multivariable averaging on sparse sets},
author = {Patrick LaVictoire and Andrew Parrish and Joseph Rosenblatt},
journal= {arXiv preprint arXiv:1206.4381},
year = {2012}
}