The Minimum Period of the Ehrhart Quasi-polynomial of a Rational Polytope

If $P\subset {\mathbb{R}}^d$ is a rational polytope, then i_P(n):=#(nP\cap \Z^d) is a quasi-polynomial in $n$, called the Ehrhart quasi-polynomial of $P$. The period of $i_P(n)$ must divide $\LL(P)= \min {n \in \Z_{> 0} \colon nP \text{is an integral polytope}}$\mathrm{.}Fewexamplesareknownwheretheperiodisnotexactly$\LL(P)$\mathrm{.}Weshowthatforany$\LL$,thereisa2-dimensionaltriangle$P$suchthat$\LL(P)=\LL$butsuchthattheperiodof$i_P(n)$is1,thatis,$i_P(n)$isapolynomialin$n$\mathrm{.}Wealsocharacterizeallpolygons$P$ such that $i_P(n)$ is a polynomial. In addition, we provide a counterexample to a conjecture by T. Zaslavsky about the periods of the coefficients of the Ehrhart quasi-polynomial.