Circling the uniform distribution

Abstract

Distributions modulo one of slowly varying sequences are of wide interest in analysis and number theory. For every real sequence (xn) for which limn→∞⁡n(xn+1−xn) exists, this article describes the precise asymptotics of the associated empirical distributions modulo one, utilizing the Kantorovich metric for probability measures on the circle. It is shown in a new, quantitative way that all limit points of these empirical distributions have the same distance from the uniform distribution and form a continuum homeomorphic to a circle. Sharp rates of convergence to that circle are identified for several classes of examples. The results strengthen and complement known facts in the literature. Appropriately adjusted, the main conclusions turn out to be valid also for the sequence (blog⁡pn) where b is a real constant and pn denotes the n-th prime number.