On the probability of a random lattice avoiding a large convex set

Given a set C in R^d, let p(C) be the probability that a random d-dimensional unimodular lattice, chosen according to Haar measure on SL(d,Z)\SL(d,R), is disjoint from C\{0}. For special convex sets C we prove bounds on p(C) which are sharp up to a scaling of C by a constant. We also prove bounds on a variant of p(C) where the probability is conditioned on the random lattice containing a fixed given point p. Our bounds have applications, among other things, to the asymptotic properties of the collision kernel of the periodic Lorentz gas in the Boltzmann-Grad limit, in arbitrary dimension d.