The central limit theorem for Riesz-Raikov sums

Summary Letf be a real-valued function with period 1 satisfying some regularity conditions. It will be proved that, for any₁,...,_d > 1, the distribution of on the probability space ([0, 1],dt) converges to a normal distribution whose covariance is given by algebraic relations among the_i's. This generalizes the classical work by M. Kac and refines the characterization off to have a degenerate limit. It also shows that the limit law of is in most cases a Cauchy distribution.