Ehrhart Quasi-Polynomials of Almost Integral Polytopes

A lattice polytope translated by a rational vector is called an almost integral polytope. In this paper, we study Ehrhart quasi-polynomials of almost integral polytopes. We study the connection between the shape of polytopes and the 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.