  It is generally hard to count, or even estimate, how many integer points lie
in a polytope P. Barvinok and Hartigan have approached the problem by way of
information theory, showing how to efficiently compute a random vector which
samples the integer points of P with (computable) constant mass, but which may
also land outside P. Thus, to count the integer points of P, it suffices to
determine the frequency with which the random vector falls in P.
  We prove a collection of efficiently computable upper bounds on this
frequency. We also show that if P is suitably presented by n linear
inequalities and m linear equations (m fixed), then under mild conditions
separating the expected value of the above random vector from the origin, the
frequency with which it falls in P is O(n^{-m/2}) as n -> infinity. As in the
classical Littlewood-Offord problem, all results in the paper are obtained by
bounding the point concentration of a sum of independent random variables; we
sketch connections to previous work on the subject.
