Quenched and annealed temporal limit theorems for circle rotations

Let h(x) = {x} − 12 . We study the distribution of ∑n−1 k=0 h(x+ kα) when x is fixed, and n is sampled randomly uniformly in {1, . . . , N}, as N → ∞. Beck proved in [Bec10, Bec11] that if x = 0 and α is a quadratic irrational, then these distributions converge, after proper scaling, to the Gaussian distribution. We show that the set of α where a distributional scaling limit exists has Lebesgue measure zero, but that the following annealed limit theorem holds: Let (α, n) be chosen randomly uniformly in R/Z× {1, . . . , N}, then the distribution of ∑n−1 k=0 h(kα) converges after proper scaling as N →∞ to the Cauchy distribution.