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Poisson statistics via the Chinese remainder theorem

A. Granville·P. Kurlberg
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Abstract

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/q_1q_2Z that are created via the Chinese remainder theorem from subsets of Z/q_1Z and Z/q_2Z (for q_1,q_2 coprime), and give criteria for when the spacings modulo q_1q_2 are Poisson. We also give some examples when the spacings modulo q_1q_2 are not Poisson, even though the spacings modulo q_1 and modulo q_2 are both Poisson.

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A. Granville, P. Kurlberg. 2005-02-21. Poisson statistics via the Chinese remainder theorem. https://arxiv.org/abs/math/0412135

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