### Abstract

It was shown by S. Kalikow and B.Weiss that, given a measure-preserving action of Zd on a probability space X and a nonnegative measurable function f on X, the probability that the sequence of ergodic averages 1/(2k + 1)d Σ gϵ[-k,...,k]d f(g - x) has at least n fluctuations across an interval (α β) can be bounded from above by c_{1}c^{n} _{2} for some universal constants c1 ϵ R and c2 ϵ (0, 1), which depend only on d; α β. 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.

Original language | English |
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Pages (from-to) | 255-273 |

Number of pages | 19 |

Journal | Studia Mathematica |

Volume | 240 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2018 |

### Keywords

- Fluctuations of ergodic averages
- Groups of polynomial growth

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## Cite this

*Studia Mathematica*,

*240*(3), 255-273. https://doi.org/10.4064/sm8692-5-2017