On the CLT for rotations and BV functions

Abstract: Let $x \mapsto x+ \alpha$ be a rotation on the circle and let $\varphi$ be a step function. We denote by $\varphi_n (x)$ the corresponding ergodic sums $\sum_{j=0}^{n-1} \varphi(x+j \alpha)$. Under an assumption on $\alpha$, for example when $\alpha$ has bounded partial quotients, and a Diophantine condition on the discontinuity points of $\varphi$, we show that $\varphi_n/|\varphi_n|_2$ is asymptotically Gaussian for $n$ in a set of density 1. The method is based on decorrelation inequalities for the ergodic sums taken at times $q_k$, where the $q_k$'s are the denominators of $\alpha$.