On the Number of Integer Points in Translated and Expanded Polyhedra

We prove that the problem of minimizing the number of integer points in parallel translations of a rational convex polytope in ℝ^6 is NP-hard. We apply this result to show that given a rational convex polytope P⊂ℝ^6 , finding the largest integer t s.t. the expansion tP contains fewer than k integer points is also NP-hard. We conclude that the Ehrhart quasi-polynomials of rational polytopes can have arbitrary fluctuations.