Super-Uniformity of The Typical Billiard Path

Abstract

1. Introduction. Uniform distribution—also called equidistribution—is a truly fundamental concept. The famous Weyl’s Criterion (about uniform distribution) is perhaps the most frequently quoted result in whole mathematics. This volume celebrates the 70th birthday of Endre Szemerédi: the famous Szemerédi’s Regularity Lemma is also about uniform distribution. It states, very roughly speaking, that every dense graph can be decomposed into a few “regular” parts, where the parts exhibit some kind of local edgeequidistribution. This paper is basically about the quantitative aspects of the continuous Kronecker-Weyl equidistribution theorem.