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Fluctuations of ergodic averages for actions of groups of polynomial growth

Volume 240 / 2018

Nikita Moriakov Studia Mathematica 240 (2018), 255-273 MSC: Primary 28D05, 28D15. DOI: 10.4064/sm8692-5-2017 Published online: 9 October 2017

Abstract

It was shown by S. Kalikow and B. Weiss that, given a measure-preserving action of $\mathbb {Z}^d$ on a probability space $\mathrm {X}$ and a nonnegative measurable function $f$ on $\mathrm {X}$, the probability that the sequence of ergodic averages $$ \frac 1 {(2k+1)^d} \sum _{g \in [-k,\dots ,k]^d} f(g \cdot x) $$ has at least $n$ fluctuations across an interval $(\alpha ,\beta )$ can be bounded from above by $c_1 c_2^n$ for some universal constants $c_1 \in \mathbb {R}$ and $c_2 \in (0,1)$, which depend only on $d,\alpha ,\beta $. The purpose of this article is to generalize this result to measure-preserving actions of groups of polynomial growth. As the main tool we develop a generalization of the effective Vitali covering theorem to groups of polynomial growth.

Authors

  • Nikita MoriakovDelft Institute of Applied Mathematics
    Delft University of Technology
    P.O. Box 5031
    2600 GA Delft, The Netherlands
    e-mail

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