Maximal periods of (Ehrhart) quasi-polynomials

A quasi-polynomial is a function defined of the form q(k) = cd(k) k d + cd−1(k) k d−1 + · · · + c0(k), where c0, c1, . . . , cd are periodic functions in k ∈ Z. Prominent examples of quasipolynomials appear in Ehrhart’s theory as integer-point counting functions for rational polytopes, and McMullen gives upper bounds for the periods of the cj(k) for Ehrhart quasi-polynomials. For generic polytopes, McMullen’s bounds seem to be sharp, but sometimes smaller periods exist. We prove that the second leading coefficient of an Ehrhart quasi-polynomial always has maximal expected period and present a general theorem that yields maximal periods for the coefficients of certain quasi-polynomials. We present a construction for (Ehrhart) quasi-polynomials that exhibit maximal period behavior and use it to answer a question of Zaslavsky on convolutions of quasipolynomials.