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Unformatted text preview: A THEOREM OF MINKOWSKI; THE FOUR SQUARES THEOREM PETE L. CLARK 1. Minkowkski’s Convex Body Theorem 1.1. Introduction. 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 r Ω of Ω by positive integers r . We found that for any “reasonable” Ω, (1) L Ω ( r ) := #( r Ω ∩ 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 PQ = { (1 − t ) P + tQ | ≤ t ≤ 1 } is contained in Ω. A subset Ω ⊂ R N is centrally symmetric if whenever it con- tains 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 . Thanks to Laura Nunley and Daniel Smitherman for finding typos. 1 2 PETE L. CLARK v) If Ω is a convex body, then ∃ P ∈ Ω; then − P ∈ Ω and 0 = 1 2 P + 1 2 ( − P ) ∈ Ω. vi) The open and closed balls of radius r with center P are convex bodies iff P = 0. Warning: The term “convex body” often has a similar but slightly different mean- ing: e.g., according to Wikipedia, a convex body is a closed, bounded convex subset Ω of R N which has nonempty interior (i.e., there exists at least one point P of Ω such that for sufficiently small ϵ > 0 the entire open ball B ϵ ( P ) of points of R N of distance less than ϵ from P is contained in Ω). Our definition of convex body is chosen so as to make the statement of Minkowski’s Theorem as clean as possible....
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This note was uploaded on 10/26/2011 for the course MATH 4400 taught by Professor Staff during the Spring '11 term at UGA.

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