On the CLT for rotations and BV functions

Abstract

Let $x\mapsto x+\alpha$ be a rotation on the circle and let $\phi$ be a step function. We denote by $\phi \_n(x)$ the corresponding ergodic sums $\sum \_{j=0}^{n-1}\phi (x+j\alpha )$. Under an assumption on $\alpha$, for example when $\alpha$ has bounded partial quotients, and a Diophantine condition on the discontinuity points of $\phi$, we show that $\phi \_n/\Vert \phi \_n\Vert \_2$ 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 $\alpha$.