On the CLT for rotations and BV functions

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

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