Aled Walker, CRM
Title: Pair correlations of fractional parts
Abstract: Given a set of natural numbers A, and a real number $alpha$, studying the distribution of the set $alpha A$ modulo 1 has been a central theme in analytic number theory for over 100 years. One has the classical equidistribution theory of Weyl, but this talk will be focused instead on the existence (or otherwise) of limiting distributions for the gap-lengths between nearby elements of the set $alpha A$ modulo 1. In certain cases, such as when $A = {1,dots, N}$, these gap-lengths are very well understood (in this instance by three-gap theorem of S贸s and 艢wierczkowski from the 50s). But what can be said in the case when A is the set of the first N squares, or k^{th} powers, or primes, or a more general sequence?