Counting lattice points interior to a polygon

Question

If I define an integer lattice $\Lambda \subseteq \mathbb{Z}^2$ with a basis given by $$\omega_{1} = a \hat{i} + b\hat{j}, \;\;\; \omega_{2} = -b \hat{i} + a\hat{j}$$ How can I count how many lattice points are interior to the fundamental domain of the lattice $\Lambda$? That is, how can I count the number of lattice points interior to the square which touches the origin in terms of $a,b$? I suspect that it is $a^2 + b^2 -1$ from drawing out a few pictures, but I'm not sure how to prove it. More generally, if I have an orthogonal basis for an integer lattice in $\mathbb{Z}^{n}$, how can I count the number of lattice points inside the $n$-dimensional hypercube which touches the origin? I would be happy with understanding the case of $\mathbb{Z}^2$ though.

Answer 1

From Pick's Theorem at Wikipedia we have $$ \mbox{interior} = \mbox{Area} + 1 - \frac{\mbox{boundary}}{2} $$

As long as your $\gcd(a,b) = 1,$ the only lattice points on the boundary of the square are the four vertices themselves. And your area is $a^2 + b^2$ in any case, so $$ \mbox{interior} = a^2 + b^2 + 1 - \frac{4}{2} $$

Next, see what happens if you allow $g = \gcd(a,b) > 1.$ You get $a^2 + b^2 + 1 - 2 g.$ Go figure.

Answer 2

Take a look at my paper at http://www.kurims.kyoto-u.ac.jp/EMIS/journals/AMUC/_vol-82/_no_1/_ionascu/ionascu.pdf

Proposition 1.1 on page 151. In general, you need to look at the literature on Ehrhart's polynomial.