Consider a $d$-dimensional convex rational polytope $P\subset\mathbb{Q}^d\subset\mathbb{R}^d$. Then, it's a standard fact that in general the function counting the number of lattice points inside the multiples $t\cdot P$ of $P$ is a quasi-polynomial instead of a polynomial, as in the integral case. This is the so-called Ehrhart quasi-polynomial of $P$.

This means that if $L(t,P)=\#\{x\in\mathbb{Z}^d:x\in t\cdot P\}$, then there exists a finite number of polynomial functions $f_1,\dots, f_D$, all of degree $\dim P$, such that $L(t,P)=f_i(t)$ whenever $t\equiv i\mod D$ (n.b. $D$ is the period of the polytope and if the polytope is integral we have $D=1$ and recover the usual Ehrhart polynomial of $P$).

What I want to know is how much can these polynomial functions differ from one another. That is, are there known bounds to the order of $f_i-f_j$ for all $i$ and $j$? How sharp are these bounds?

The case I'm dealing with is for $\dim P=2$, so if there are sharper results for this case than in general that'd be great.

constituent. $\endgroup$ – matthias beck May 28 '18 at 4:47