Random walks on the circle and Diophantine approximation

Random walks on the circle group (Formula presented.) whose elementary steps are lattice variables with span (Formula presented.) or (Formula presented.) taken mod (Formula presented.) exhibit delicate behavior. In the rational case, we have a random walk on the finite cyclic subgroup (Formula presented.), and the central limit theorem and the law of the iterated logarithm follow from classical results on finite state space Markov chains. In this paper, we extend these results to random walks with irrational span (Formula presented.), and explicitly describe the transition of these Markov chains from finite to general state space as (Formula presented.) along the sequence of best rational approximations. We also consider the rate of weak convergence to the stationary distribution in the Kolmogorov metric, and in the rational case observe a phase transition from polynomial to exponential decay after (Formula presented.) steps. This seems to be a new phenomenon in the theory of random walks on compact groups. In contrast, the rate of weak convergence to the stationary distribution in the total variation metric is purely exponential.