Let $N$ be a positive integer and let $P \subset \mathbb{C}$ be a polygon whose vertices are of the form $(a_1+b_1 i)/N$, $(a_2+b_2 i)/N$, ..., $(a_r+b_r i)/N$, with $a_j + b_j i$ being various Gaussian integers. For any Gaussian integer $u+vi$, let $h_P(u+vi)$ be the number of lattice point in $(u+vi) \cdot P$. I would like to say that $h_P(u+vi)$ is in some sense "quasi-polynomial modulo $N$".
The case $N=1$ can be studied using [Pick's theorem][1]. The area of $P$ transforms simply under multiplication by Gaussian integers, the number of lattice points on the boundary is only a little messier. In particular, it is easy to show that there is some non-zero Gaussian integer $D$ such that, as long as $GCD(D, u+vi)=1$ and $GCD(u,v)=1$, then $h_P(u+vi) = (u^2+v^2) \mathrm{Area}(P) + \mbox{constant}$.
A result that would satisfy me is that there is some non-zero non-zero Gaussian integer $D$ such that, as long as $GCD(D, u+vi)=1$ and $GCD(u,v)=1$, then
$$h_p(u,v) = (u^2+v^2) \mathrm{Area}(P) + a(u,v) u + b(u,v) v + c(u,v)$$
where $a$, $b$ and $c$ are periodic modulo $N$ in each input.
Motivation: This is enough to give an elementary deduction that the [quartic residue][2] symbol $\left[ \frac{z}{u+vi} \right]$ is periodic as a function of $u+vi$.
Motivation behind the motivation: There are a series of papers: [Habicht][3], [Kubota][4], [Hill][5] which promise to prove $m$-th power reciprocity laws roughly by counting lattice points. I can't follow any of them, so I thought I would true to reverse engineer what might be true in order to make the $\mathbb{Z}[i]$ proof come out.
For this reason, I would also be interested in statements for other number fields, but I don't have a precise question worked out in that setting.
[1]: http://en.wikipedia.org/wiki/Pick%27s_theorem
[2]: http://en.wikipedia.org/wiki/Quartic_reciprocity#Quartic_residue_character
[3]: http://www.ams.org/mathscinet-getitem?mr=113867
[4]: http://www.ams.org/mathscinet-getitem?mr=921585
[5]: http://www.ams.org/mathscinet-getitem?mr=1324545