Spacing Statistics for Sequences modulo 1

Abstract

The thesis aims at exploring the spacing statistics of sequences modulo one. We focus particularly on uniform distribution, correlation statistics, and level-spacing (gap) distribution of sequences modulo 1. Uniform distribution has a rich history of its own, which is said to have begun with Hermann Weyl in 1916. With the development of correlation statistics, particularly pair correlation, people began to explore the relation between these two notions. We study this relation between uniform distribution and correlation statistics, especially when the latter is Poissonian. We aim to gain further insight by collecting several examples spread through the literature, and observing the spacing statistics they possess. There has also been an emergence of smooth analogues of these statistics, and we explore the interplay between the classical and the smooth analogues. In particular, we show that the smooth analogues imply the classical definitions in the case of Poissonian statistics. We further try to derive a criterion for the existence of the Poissonian pair correlation of a sequence modulo 1. It was shown by Kurlberg and Rudnick in 1999 that if a sequence admits the Poissonian correlations of all orders, then we can recover the level-spacing distribution function of the sequence, and it turns out to be Poissonian as well. We keenly study this argument, and fill in some details to improve its readability.