Quenched and annealed temporal limit theorems for circle rotation

Let h(x) = {x} - 1/2. We study the distribution of Sigma(n-1)(k=0) h(x + k alpha) when x is fixed, and n is sampled randomly uniformly in {1, ..., N}, as N -> infinity. Beck proved in [2, 3] that if x = 0 and alpha is a quadratic irrational, then these distributions converge, after proper scaling, to the Gaussian distribution. We show that the set of alpha where a distributional scaling limit exists has Lebesgue measure zero, but that the following annealed limit theorem holds: Let (alpha, n) be chosen randomly uniformly in R/Z x {1, ..., N}, then the distribution of Sigma(n-1)(k=0) h(k alpha) converges after proper scaling as N -> infinity to the Cauchy distribution.