Asymptotic distributions for normalized ergodic sums over rotations

Let $x \to x+ \alpha$ be a rotation on the circle and let $\varphi$ be a function with bounded variation. Denote by $S_n(\varphi, x) := \sum_{j=0}^{n-1} \varphi(x+j \alpha)$ the ergodic sums.

For a large class of $\alpha$’s including irrationals with bounded partial quotients, we show decorrelation inequalities between the ergodic sums at time $q_k$, where the $q_k$’s are the denominators of $\alpha$.

This allows to study the asymptotic distribution of the ergodic sums $S_n(\varphi, x)$ after normalization, in particular for some step functions, along subsequences.

We will give also an application to a geometric model, the billiard flow in the plane with periodic rectangular obstacles when the flow is restricted to special directions.