On the CLT for rotations and BV functions

Let $x?x+\alpha$ be a rotation on the circle and let φ be a step function. Denote by ${\phi }_n(x)$ the ergodic sums ${\sum }_{j=0}^{n?1}\phi (x+j\alpha )$. For α in a class containing the rotations with bounded partial quotients and under a Diophantine condition on the discontinuities of φ, we show that ${\phi }_n/{6{\phi }_{n}6}_2$ is asymptotically Gaussian for n in a set of density 1.