Poisson statistics via the Chinese remainder theorem

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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 have 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. 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.