Maximal periods of (Ehrhart) quasi-polynomials

A quasi-polynomial is a function defined of the form q ( k ) = c d ( k ) k d + c d − 1 ( k ) k d − 1 + ⋯ + c 0 ( k ) , where c 0 , c 1 , … , c d are periodic functions in k ∈ Z . Prominent examples of quasi-polynomials appear in Ehrhartʹs theory as integer-point counting functions for rational polytopes, and McMullen gives upper bounds for the periods of the c j ( k ) for Ehrhart quasi-polynomials. For generic polytopes, McMullenʹs 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 quasi-polynomials.