background

# Applications to Minkowski's Theorem

Theorem 1   Any convex set (or body) in that has central symmetry and volume greater than contains an integer lattice point other than .

If is an invertible matrix, then is a linear map. The lattice is mapped into a system of points in which we call a lattice , where is equal to the determinant of the lattice .

Theorem 2   Let be any lattice of determinant . Then any convex set symmetrical about the origin whose volume is greater than contains a point of other than .

Example 3   Given any real number and integer , there exists integers such that .

Take -set parallelogram bounded by the four lines

where .

This parallelogram has base , altitude , hence . If we take , then . By Minkowski's fundamental theorem, there must be at least one lattice point other than . Thus

or,

Recall , so

QED

Example 4   Lagrange: Every positive integer can be expressed as a sum of four squares,

where are non-negative integers.

In view of the algebraic identity

we see that are sums of four squares, thus is the sum of four squares. Therefore, it is enough to prove the theorem for primes. Let

where are chosen so that    mod. Let and suppose that is a point of . If , then
 mod mod   .

So we have . Now consider the ball

Thus, is convex and symmetric about 0. A ball of radius has volume . If we take , we see
 by Minkowski's Fundamental Theorem mod

so it has to be !

QED

Theorem 5   There exists infinitely many positive integers that can be written as a sum of four positive perfect squares, but every integer is a sum of five positive perfect squares.

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