We present a proof of a famous theorem of Hermann Minkowski which spawned a beautiful branch of number theory known as the Geometry of Numbers. This theorem gives a condition on the volume of a centrally symmetric convex body in -dimensional Euclidean space to contain at least one lattice point except the trivial point .
Definitions and Notations
Definition 1 In an -dimensional vector space over the field , a lattice is a free abelian subgroup of of the following form
where are linearly independent vectors in . Then is a free abelian group of rank . A lattice is said to be complete if . Assuming to be complete, we define the fundamental mesh in with respect the the to be
The volume of the lattice is defined to be the volume of the fundamental mesh denoted as , i.e. . There are a few equivalent expressions for this volume. One expression convenient for computations is given by the following relation
where is the usual inner-product (or a symmetric positive definite bilinear form) defined on the vector space .
Definition 2 (Convex and Centrally Symmetric Body)
A subset of is said to be centrally symmetric if for any point we have , and the subset is convex if for any two distinct points in the line segment is contained in . For example, a ball is convex in but a solid torus is not.
Minkowski’s Lattice Point Theorem
Theorem: Let be an -dimensional Euclidean vector space (-vector space) and be a complete lattice of . Let be a centrally symmetric convex subset of such that
Then contains at least one point other than in .
Proof: Consider the dilation . For any , consider the translated sets . We will show that there exist such that and
For the sake of contradiction, assume the sets are pairwise disjoint. Then the intersections are also disjoint for a fundamental mesh . This gives us the following inequality involving volumes
We know that translation preserves volumes, so
Hence for each we have
We claim that the cover as varies over . Let . Since is a set of linearly independent vectors in and , it follows that is a basis of . Then such that . We know that every real number can be written as where and . Then
Note that and . Taking we observe that . Hence, . Since for all we have . Hence , and so the sets cover . Therefore, we finally have
which is a contradiction to our initial assumption since . Hence, we can choose such that
Therefore, there exist , such that
Since is centrally symmetric and convex, and thus Therefore and . This completes the proof of the theorem.
Since we also have and . This means contains at least two distinct lattice points. What else can you find hidden in the proof?
A Useful Corollary
Taking and , we observe that any convex centrally symmetric body in of volume strictly bigger that contains at least one point with integer coordinates other than .
Minkowski’s Theorem in Action
Now we present a number theoretic application of Minkowski’s theorem. The following result was proved by Axel Thue using the pigeonhole principle. We give a proof using Minkowski’s lattice point theorem.
Theorem: Primes of the form can be expressed as a sum of two squares.
Proof. Let be a prime of the form . Then is a quadratic residue modulo or, equivalently, there exists such that . Consider the two vectors and in . Let for some . This gives us and hence . Therefore are linearly independent. Then is a complete lattice in with .
Let . There exist such that . This implies . Hence,
Consider the open disc of radius centered at the origin . We have,
Thus, is convex and centrally symmetric. By Minkowski’s theorem, there exists a lattice point apart from the origin in . Let this point be . Then and and hence . We are done!
 Andreescu, T. and Dospinescu, G., 2008. Problems from the Book.
 Neukirch, J., 2013. Algebraic number theory (Vol. 322). Springer Science & Business Media.
 Cassels, J.W.S., 2012. An introduction to the geometry of numbers. Springer Science & Business Media.