  A lattice polytope translated by a rational vector is called an almost
integral polytope. In this paper we investigate Ehrhart quasi-polynomials of
almost integral polytopes. We study the relationship between the shape of the
polytopes and algebraic properties of the Ehrhart quasi-polynomials. In
particular, we prove that lattice zonotopes and centrally symmetric lattice
polytopes are characterized by Ehrhart quasi-polynomials of their rational
translations.
