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arXiv · 1206.4381

Multivariable averaging on sparse sets

Patrick LaVictoire·Andrew Parrish·Joseph Rosenblatt
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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.

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Patrick LaVictoire, Andrew Parrish, Joseph Rosenblatt. 2012-06-20. Multivariable averaging on sparse sets. https://arxiv.org/abs/1206.4381

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