# Fluctuations of ergodic averages for amenable group actions

## TL;DR

This paper establishes exponential decay estimates for the probability of fluctuations in ergodic averages for amenable group actions, extending previous results to a broader class of groups and F{46}lner sequences.

## Contribution

It introduces a universal estimate for fluctuations in ergodic averages along specific F{46}lner sequences for amenable groups, generalizing prior work.

## Key findings

- Provides exponential decay bounds for fluctuation probabilities.
- Shows any countable amenable group admits suitable F{46}lner sequences.
- Extends classical results from 46}lner sequences in 46}Z^d to general amenable groups.

## Abstract

We show that for any countable amenable group action, along F{\o}lner sequences that have for any $c>1$ a two sided $c$-tempered tail, one have universal estimate for the probability that there are $n$ fluctuations in the ergodic averages of $L^{\infty}$ functions, and this estimate gives exponential decay in $n$. Any two-sided F{\o}lner sequence can be thinned out to satisfy the above property, and in particular, any countable amenble group admits such a sequence. This extends results of S. Kalikow and B. Weiss for $\mathbb{Z}^{d}$ actions and of N. Moriakov for actions of groups with polynomial growth.