On the number of integer points in translated and expanded polyhedra

We prove that the problem of minimizing the number of integer points inparallel translations of a rational convex polytope in ${\mathbb{R}}^6$ is NP-hard. We apply this result to show that given a rational convex polytope $P \subset $\mathbb{R}$^6$,findingthelargestinteger$t$s\mathrm{.}t\mathrm{.}theexpansion$tP$containsfewerthan$k$ integer points is also NP-hard. We conclude that the Ehrhart quasi-polynomials of rational polytopes can have arbitrary fluctuations.