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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

Multivariable averaging on sparse sets


Authors: P. LaVictoire, A. Parrish and J. Rosenblatt
Journal: Trans. Amer. Math. Soc. 366 (2014), 2975-3025
MSC (2010): Primary 37A45
Published electronically: February 6, 2014
MathSciNet review: 3180737
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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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Additional Information

P. LaVictoire
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Address at time of publication: Principal Engineer, Quixey, 278 Castro Street, Mountain View, California 94041
Email: patlavic@math.wisc.edu, patrick@quixey.com

A. Parrish
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
Email: ajnparrish@gmail.com

J. Rosenblatt
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
Email: rosnbltt@illinois.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2014-06084-4
Received by editor(s): June 21, 2012
Published electronically: February 6, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.