How different can the constituents of an Ehrhart quasi-polynomial be? 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.
 A: Let $d=\dim(P)$.  First, since $L(t,P)$ is non-decreasing in $t$, for any positive integer $n$ we have 
$$f_i((n-1)D+i) \leq f_j((n-1)D+j) \leq f_i(nD+i) \leq f_j(nD+j)$$
whenever $i \leq j$.  Thus it is easy to see that the coefficients of $t^d$ in $f_i,f_j$ must be the same (in fact this coefficient is a normalized volume of $P$).  Thus $f_i(t)-f_j(t)$ is $O(t^{d-1})$.
It is easy to see that $\Omega(t^{d-1})$ can be achieved.  For example, if $P$ is the $d$-cube with side-length $1/2$ in the positive quadrant with a vertex at the origin, then 
$$f_1(t)=\left(\frac{1}{2}\right)^d (t+1)^d$$ and 
$$f_2(t)=\left(\frac{1}{2}\right)^d (t+2)^d$$
so the difference is at least $\frac{d}{2^d}t^{d-1}$.  Probably one can improve this constant.
A: I'll complement Christian's answer with an example in the other direction. Consider the polytope of $8\times 8$ symmetric doubly-stochastic matrices with 0 diagonal. The period of the Ehrhart quasipolymonial is 2 and the degree is 20.  However the difference between the polynomials for even and odd dilations has degree only 5.  I don't know (but would like to know) what happens for larger matrices.
