Poisson Approximation of Random Lattices

Let S in R^n be a measurable set of volume at most n/5 such that -S is disjoint from S. We consider two point processes in S: the first is a Poisson point process of intensity one, and the second is the restriction of a random lattice L to S, where L is distributed uniformly in the space of covolume-one lattices. Building upon old works by Rogers, Schmidt and others, we prove that the total variation distance between these two point processes is at most C e^{-c’ n}.