2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/135399We 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 $β\mapsto \set{α+β}$, $α\in \R\setminus \Q$. In particular we obtain sharp results for the diffusion of the walk on $\Z$ generated by the location of points of the sequence $\{nα+β\}$ on a binary partition of the unit interval. Finally we give some applications of our method.27 pagesDynamical SystemsNumber Theory37E05, 37A25, 37A45text