On the CLT for rotations and BV functions

Let $x\mapsto x+\alpha$ be a rotation on the circle and let $\phi$ be a step function. We denote by $\phi \mathrm{\_}n(x)$ the corresponding ergodic sums $\sum \mathrm{\_}{j=0}^{n-1}\phi (x+j\alpha )$. 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 $\phi \mathrm{\_}n\mathrm{/}\mathrm{\parallel }\phi \mathrm{\_}n\mathrm{\parallel }\mathrm{\_}2$ 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 $q\mathrm{\_}k$, where the $q\mathrm{\_}k$’s are the denominators of $\alpha$.