  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.
