# Fast and slow points of Birkhoff sums

## TL;DR

This paper studies the growth behavior of Birkhoff sums for continuous zero-mean functions on the circle, revealing how typicality assumptions influence the growth rates in dynamical systems.

## Contribution

It provides a detailed analysis of the growth rates of Birkhoff sums under different notions of typicality, extending the understanding to more general contexts.

## Key findings

- Growth rates depend on the interpretation of 'typical'
- Analysis applies to continuous functions with zero mean
- Results extend to broader dynamical systems contexts

## Abstract

We investigate the growth rate of the Birkhoff sums $S_{n,\alpha} f(x)=\sum_{k=0}^{n-1} f(x+k\alpha)$, where $f$ is a continuous function with zero mean defined on the unit circle $\mathbb T$ and $(\alpha,x)$ is a "typical" element of $\mathbb T^2$. The answer depends on the meaning given to the word "typical". Part of the work will be done in a more general context.