Aled Walker, Cambridge
In the late 1990s, Rudnick and Sarnak conjectured that the gap distribution of the sequence of dilated squares modulo 1, at least for a generic dilate, should agree with the gap distribution of a set of uniformly distributed random points modulo 1. This conjecture is still completely open. Nonetheless, the conjecture has stimulated a great deal of work, studying these gap distributions of dilated squares and dilates of other sequences, particularly focussed on the associated correlation functions. Recently, connections were discovered to certain notions from additive combinatorics and sum-product theory. In this talk I will discuss some of the work I’ve been involved with on pair correlations and triple correlations related to these problems, studying dilates of the primes, the squares, and of generic sequences — sometimes jointly with various subsets of Thomas Bloom, Sam Chow, Ayla Gafni, and Niclas Technau.