Covering Numbers: Arithmetics and Dynamics for Rotations and Interval Exchanges

We study a particular case of the two-dimensional Steinhaus theorem , giving estimates of the possible distances between points of the form k and k + on the unit circle, through an approximation algorithm of by the points k. This allows us to compute covering numbers (maximal measure of Rokhlin stacks having some prescribed regularity properties) for the symbolic dynamical systems associated to the rotation of argument acting on the partition of the circle by the points 0,. We can then compute topological and measure-theoretic covering numbers for exchange of three intervals; in this way, we prove that every ergodic exchange of three intervals has simple spectrum, and build a new class of three-interval exchanges which are not of rank one.