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}<e_1,e_2,...,e_n>. 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)<vol(C). 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.


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