# the  geometry of numbers

The geometry of numbers is q branch of number theory created by Minkowski.

The idea of this theory is to consider the intersection of some set with a lattice in the Euclidean space $\mathbb{R}^n$ to get the existence or non-existence of some particular number.

Start with the basics. Suppose $L$ is a lattice in $\mathbb{R}^n$(i.e. a discret subgroup of the additive group of $\mathbb{R}^n$ without point of accumulation, which viewed as a set of vectors in $\mathbb{R}^n$ generate the whole space). Using the fact that $\mathbb{R}^n$ has a finite set as basis, we deduce easily that $L$ is finitely generated. Using the fact that $\mathbb{Z}$ is a principal domain, we see that $L$ has, in fact, a finite set as $\mathbb{Z}$-basis(what is more, the cardinality of this set is independent of the choice of the basis, it is an invariant under isomorphism, also a property of the abelian group). That is we can assume that $L$ looks like $L=\mathbb{Z}$. What is more, we can for any other basis $\{f_1,f_2,...,f_n\}$ of $L$, the matrix of these $f_i$ expressed with respect to these $e_j$ lies in the group $GL_n(\mathbb{Z})$. After choosing the canonical basis of $\mathbb{R}^n$, we can calculate the determinant of the matrix $(e_1,...,e_n)$. The absolute value of this determinant is, coincidently, the volume of the fundamental domain of the lattice $L$, or in other words, the volume of $\mathbb{R}^n/L$ equipped with the quotient measure(we denote it by $covol(L)$, the co-volume of $L$). This value is, in some sense, the generator of the image of the homomorphism of groups $f:L^n\rightarrow \mathbb{R}, (a_1,...,a_n)\mapsto det(a_1,...,a_n)$.

The fundamental result in the geometry of numbers concerns the intersection of a convex set with the lattice. We say a convex subset $C\subset \mathbb{R}^n$ is symmetric if for all $x\in C$, we have $-x\in C$. So if $x,y\in C$, we have that $latex\frac{x-y}{2}$. This is really a simple fact, but it is also a really important observation. Now we are ready to state the theorem of Minkowski:

Suppose that $C$ a convex symmetric subset of $\mathbb{R}^n$, and $L$ is a lattice of the latter. What is more, $covol(L)<\frac{vol(C)}{2^n}$. Then there is a non-zero element $0\neq x\in L$ which lies in $C$ also. That is to say, $\#C\bigcap L>1$.

The point of the proof is to express the above inequality as a comparison between two volumes. Noting that if we dilate the lattice $L$ by $2$, then we have that $covol(2L)=2^ncovol(L)$. So the inequality turns into $covol(2L). Then we can say that there are two distinct elements $x\neq y\in 2L$ such that the half of their difference $\frac{x-y}{2}\in C$. If it is not the case, then after choosing a measurable representation $X$ of $\mathbb{R}^n/2L$ in $\mathbb{R}^n$, we can express $C=\bigcup_{x\in 2L}(C\bigcap(X+x))$. And so $C'=\bigcup_{x\in 2L}(C\bigcap(X+x)-x)$ has the same volume as $C$. But $C'$ is now a measurable subset of $X$(because $C$ is a measurable subset, see here), so we have $vol(C)\leq covol(2L)$, which is a contradiction to the assumption. So we get a non-zero element $\frac{x-y}{2}\in C\bigcap L$, thus proved the theorem.

Now come to the applications of this fundamental result. As a first example, we consider one theorem of Fermat’s, that is a prime number $p\equiv 1(mod 4)$ if and only if $p$ can be written as a sum of two squares of integers, $p=a^2+b^2(a,b\in\mathbb{Z})$. Note that $p\equiv 1(mod 4)$ is the same as that $-1$ is quadratic residue mod $p$. So we can find some integer $u$ such that $u^2\equiv -1(mod p)$. We will use this $u$ to construct some lattice $L$ in $\mathbb{R}^2$. That is, ze define $L=\{(a,b)\in\mathbb{Z}^2,a\equiv ub(mod p)\}$. Clearly this is a lattice(a routine way of showing this is, $p\mathbb{Z}^2\subset L\subset\mathbb{Z}^2$). What is more, the indice of $L$ in $\mathbb{Z}^2$ is $p$(just consider the group homomorphism $\phi:\mathbb{Z}^2\rightarrow\mathbb{Z}/p\mathbb{Z}, (x,y)\mapsto x-uy$, which is surjective, with kernel $ker(\phi)=L$. So we have $\#\mathbb{Z}/L=p$).And thus $covol(L)=p*covol(\mathbb{Z}^2)=p$. Now consider the disk $D=\{(x,y)\in\mathbb{R}^2, x^2+y^2<2p\}$. This disk is clearly convex and symmetric, with volume $vol(D)=2p\pi$. Now that the assumptions are satisfied just by noting that $p<\frac{2p\pi}{2^2}$, so there is $0\neq x=(a,b)\in L\bigcap D$. Note that since $a=ub(mod p)$, we have $a^2=u^2b^2=-b^2(mod p)$, in other words, $a^2+b^2=np$ for some integer $n$.But recall the definition of $D$, we have that $a^2+b^2<2p$, so the only possibility is that $a^2+b^2=p$.

So by carefully choosing a lattice, we proved the theorem due to Fermat. This is a small success of the geometry of numbers. In fact, the above reasoning can be generalized to any polynomials like $a^2+db^2$ where $d$ is a positive integer.

The following is one such result:

Suppose that integer $d>0$ is a quadratic residue mod $p$, then at least one of the numbers $p,2p,..., hp$($h$ is the largest integer such that $h\leq\frac{4d}{\pi}$) can be expressed in the form $a^2+db^2$.