Poisson statistics via the Chinese Remainder Theorem

We consider the distribution of spacings between consecutive elements in subsets of Z/qZ, 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/q1q2Z that are created via the Chinese Remainder Theorem from subsets of Z/q1Z and Z/q2Z (for q1,q2 coprime), and give criteria for when the spacings modulo q1q2 are Poisson. Moreover, we also give some examples when the spacings modulo q1q2 are not Poisson, even though the spacings modulo q1 and modulo q2 are both Poisson.