Fast and slow points of Birkhoff sums

Abstract

We investigate the growth rate of the Birkhoff sums $S_{n,\unicode[STIX]{x1D6FC}}f(x)=\sum _{k=0}^{n-1}f(x+k\unicode[STIX]{x1D6FC})$, where $f$ is a continuous function with zero mean defined on the unit circle $\mathbb{T}$ and $(\unicode[STIX]{x1D6FC},x)$ 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.