We introduce a renormalization procedure which allows us to study in a unified and concise way different properties of the irrational rotations on the unit circle $\beta \mapsto \set{ \alpha+\beta}$, $\alpha \in \R\setminus \Q$. In particular we obtain sharp results for the diffusion of the walk on $\Z$ generated by the location of points of the sequence $\{ n\alpha +\beta\}$ on a binary partition of the unit interval. Finally we give some applications of our method.
Categories: math.DS math.NT