Applications to Minkowski's Theorem
Theorem 1
Any convex set (or body) in
![$ {\mathbb{R}}^n$](img1.png) that has central symmetry and volume
greater than ![$ 2^n$](img2.png) contains an integer lattice point other than
![$ \overline{0}$](img3.png) .
If is an invertible matrix, then is a linear map. The lattice
is mapped
into a system of points in
which we call a lattice , where is
equal to the determinant of the lattice .
Theorem 2
Let ![$ L$](img7.png) be any lattice ![$ L$](img7.png) of determinant
![$ \Delta\in{\mathbb{R}}^n$](img9.png) . Then any convex
set symmetrical about the origin whose volume is greater than ![$ 2^n\Delta$](img10.png) contains a point
of ![$ L$](img7.png) other than ![$ (0,0)$](img11.png) .
Example 3
Given any real number ![$ \alpha$](img12.png) and integer ![$ t'>0$](img13.png) , there exists
integers ![$ p,q$](img14.png) such that
![$ \vert\frac{p}{q}-\alpha\vert \leq \frac{1}{\vert p\vert t'}$](img15.png) .
Take -set parallelogram bounded by the four lines
where .
This parallelogram has base , altitude , hence
.
If we take
, then . By Minkowski's fundamental theorem, there
must be at least one lattice point other than . Thus
or,
Recall
, so
QED
Example 4
Lagrange: Every positive integer ![$ n$](img29.png) can be expressed as a sum of four
squares,
where ![$ x_i$](img31.png) are non-negative integers.
In view of the algebraic identity
we see that are sums of four squares, thus is the sum of four squares.
Therefore, it is enough to prove the theorem for primes. Let
where are chosen so that
mod . Let
and
suppose that is a point of . If
, then
So we have
. Now consider the ball
Thus, is convex and symmetric about 0. A ball of radius
has
volume
. If we take
, we see
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![$\displaystyle V(\zeta)$](img60.png) |
![$\displaystyle =$](img33.png) |
![$\displaystyle \frac{1}{2}\pi^2 (2p)^2$](img61.png) |
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![$\displaystyle =$](img33.png) |
![$\displaystyle 2 \pi^2 p^2$](img62.png) |
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|
![$\displaystyle >$](img63.png) |
![$\displaystyle 2^4 p^2$](img64.png) |
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![$\displaystyle \Rightarrow$](img65.png) |
by Minkowski's Fundamental Theorem ![$\displaystyle \exists \, \overline{x} \in \Lambda, \: \overline{x} \neq 0, \: \overline{x} \in \zeta$](img66.png) |
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![$\displaystyle \Rightarrow$](img65.png) |
![$\displaystyle 0<x_1^2+x_2^2+x_3^2+x_4^2<2p$](img67.png) |
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![$\displaystyle \equiv$](img50.png) |
mod![$\displaystyle \, p,$](img68.png) |
so it has to be !
QED
Theorem 5
There exists infinitely many positive integers that can be written
as a sum of four positive perfect squares, but every integer ![$ n>169$](img70.png) is a sum of
five positive perfect squares.
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