Periods of Ehrhart Coefficients of Rational Polytopes

Let P⊂R^n be a polytope whose vertices have rational coordinates. By a seminal result of E. Ehrhart, the number of integer lattice points in the kth dilate of P (k a positive integer) is a quasi-polynomial function of k — that is, a "polynomial" in which the coefficients are themselves periodic functions of k. It is an open problem to determine which quasi-polynomials are the Ehrhart quasi-polynomials of rational polytopes. As partial progress on this problem, we construct families of polytopes in which the periods of the coefficient functions take on various prescribed values.