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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.