Cyclic approximation of irrational rotations

Abstract:

We prove that an irrational number $\alpha$ admits a rational approximation $|\alpha - p/q| = o(f(q))$ iff the irrational rotation $Tx = \{ x + \alpha \}$ admits cyclic approximation with speed $o(f(n))$. As an application to earlier results we obtain that a generic Anzai skew product over every irrational rotation is rank-1 and for a.e. $\alpha$ most skew products admit cyclic approximation with speed $o(1/{n^2}\log n)$.