  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.
