Convergence rate of random walks on the circle by an irrational rotation

Abstract

Fix an irrational number α ∈ [0,1). We consider the ±α random walk on the unit circle, i.e., rotating repeatedly by an angle +2πα or -2α with each probability $\frac{1}{2}$. We investigate the convergence rate of the random walk to uniform distribution under the discrepancy. The sharp rate of the convergence is $O(k^{-\frac{1}{2}})$ if the irrational number ξ=2α has bounded partial quotients.