A Theorem of Minkowski; the Four Squares Theorem

We have already considered instances of the following type of problem: given a bounded subset Ω of Euclidean space R N , to determine #(Ω ∩ Z N), the number of integral points in Ω. It is clear however that there is no answer to the problem in this level of generality: an arbitrary Ω can have any number of lattice points whatsoever, including none at all. In [Gauss's Circle Problem], we counted lattice points not just on Ω itself but on dilates rΩ of Ω by positive integers r. We found that for any " reasonable " Ω, (1) L Ω (r) := #(rΩ ∩ Z N) ∼ r N Vol(Ω). More precisely, we showed that this holds for all bounded sets Ω which are Jordan measurable, meaning that the characteristic function 1 Ω is Riemann integrable. It is also natural to ask for sufficient conditions on a bounded subset Ω for it to have lattice points at all. One of the first results of this kind is a theorem of Minkowski, which is both beautiful in its own right and indispensably useful in the development of modern number theory (in several different ways). Before stating the theorem, we need a bit of terminology. Recall that a subset Ω ⊂ R N is convex if for all pairs of points P, Q ∈ Ω, also the entire line segment P Q = {(1 − t)P + tQ | 0 ≤ t ≤ 1} is contained in Ω. A subset Ω ⊂ R N is centrally symmetric if whenever it contains a point v ∈ R N it also contains −v, the reflection of v through the origin. A convex body is a nonempty, bounded, centrally symmetric convex set. Some simple observations and examples: i) A subset of R is convex iff it is an interval. ii) A regular polygon together with its interior is a convex subset of R 2. iii) An open or closed disk is a convex subset of R 2. iv) Similarly, an open or closed ball is a convex subset of R N .