Periods of Ehrhart coefficients of rational polytopes (Q1753034)

Summary: Let \(\mathcal{P} \subset \mathbb{R}^{n}\) be a polytope whose vertices have rational coordinates. By a seminal result of E. Ehrhart, the number of integer lattice points in the \(k\)th dilate of \(\mathcal{P}\) (\(k\) 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.