Fast and slow points of Birkhoff sums

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.