Berry–Esseen Bounds and Diophantine Approximation

Let S_N, N = 1, 2,.. be a random walk on the integers, let α be an irrational number and let Z_N = {S_Nα}, where {·} denotes fractional part. Then Z_N, N = 1, 2,.. is a random walk on the circle, and from classical results of probability theory it follows that the distribution of Z_N converges weakly to the uniform distribution. We determine the precise speed of convergence, which, in addition to the distribution of the elementary step X of the random walk S_N, depends sensitively on the rational approximation properties of α.