Multivariable averaging on sparse sets

LaVictoire, P.; Parrish, A.; Rosenblatt, J.

doi:10.1090/S0002-9947-2014-06084-4

Multivariable averaging on sparse sets
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by P. LaVictoire, A. Parrish and J. Rosenblatt PDF

Trans. Amer. Math. Soc. 366 (2014), 2975-3025 Request permission

Abstract:

Nonstandard ergodic averages can be defined for a measure-pre- serving 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 $\mathbb {Z}^d$. 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 $\mathbb {Z}^d$. The second and third results have two nontrivial variants on $\mathbb {Z}^d$: a “native” $d$-dimensional average and a “product” average from the one-dimensional averages.

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