Rational polytopes with Ehrhart coefficients of arbitrary period

A seminal result of E. Ehrhart states that the number of integer lattice points in the dilation of a rational polytope by a positive integer $k$ is a quasi-polynomial function of $k$ — that is, a “polynomial” in which the coefficients are themselves periodic functions of $k$. Using a result of F. Liu on the Ehrhart polynomials of cyclic polytopes, we construct not-necessarily-convex rational polytopes of arbitrary dimension in which the periods of the coefficient functions appearing in the Ehrhart quasi-polynomial take on arbitrary values.