Poisson statistics via the Chinese Remainder Theorem

Abstract

We consider the distribution of spacings between consecutive elements in subsets of Z / q Z , where q is highly composite and the subsets are defined via the Chinese Remainder Theorem. We give a sufficient criterion for the spacing distribution to be Poissonian as the number of prime factors of q tends to infinity, and as an application we show that the value set of a generic polynomial modulo q has Poisson spacings. We also study the spacings of subsets of Z / q 1 q 2 Z that are created via the Chinese Remainder Theorem from subsets of Z / q 1 Z and Z / q 2 Z (for q 1 , q 2 coprime), and give criteria for when the spacings modulo q 1 q 2 are Poisson. Moreover, we also give some examples when the spacings modulo q 1 q 2 are not Poisson, even though the spacings modulo q 1 and modulo q 2 are both Poisson.