Fast and slow points of Birkhoff sums

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.