Fluctuations of Ergodic Averages for Actions of Groups of Polynomial Growth

Abstract

It was shown by S. Kalikow and B. Weiss that, given a measure-preserving action of $\mathbb{Z}^d$ on a probability space $X$ and a nonnegative measurable function $f$ on $X$, the probability that the sequence of ergodic averages $$ \frac 1 {(2k+1)^d} \sum\limits_{g \in [-k,\dots,k]^d} f(g \cdot x) $$ 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 effective Vitali covering theorem for groups of polynomial growth.