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1.8: Geometry of Numbers

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We have already seen that geometrical concepts are sometimes useful in illuminating number theoretic considerations. With the introduction by Minkowski of geometry of numbers a real welding of important parts of number theory and geometry was achieved. This branch of mathematics has been in considerable vogue in the last 20 years, particularly in England where it was and is being developed vigorously by Mordell, Davenport, Mahler and their students.

We shall consider a very brief introduction to this subject. First, we shall examine a proof of the fundamental theorem of Minkowski due to Hajos (1934), then we shall discuss some generalizations and applications of this theorem, and finally we shall investigate some new results and conjectures that are closely related.

In its simplest form the fundamental theorem of Minkowski is the following.

Theorem 1.8.1: Fundamental Theorem of Minkowski

Let R be a region in the xy plane of area A>4, symmetric about the origin and convex. Then R contains a lattice point other than the origin.

First, some preliminary remarks. In the condition A>4, the 4 cannot be replaced by any smaller number. This may be seen by considering the square of side 2ϵ, centered at the origin. Indeed this example might at first suggest that the theorem is quite intuitive, as it might seem that squeezing this region in any direction and keeping its area fixed would necessarily force the region to cover some lattice point. However the matter is not quite so simple, as other examples reveal that neither central symmetry nor convexity are indispensable. As far as convexity is concerned what is really needed is that with the vectors V1 and V2 the region should also contain 12(V1+V2). The symmetry means that with V1 the vector V1 should also be in R. Thus the symmetry and convexity together imply that, if V1 and V2 are in R, so is 12(V1V2). This last condition is really sufficient for our purpose and may replace the conditions of symmetry and convexity. It is implied by symmetry and convexity but does not imply either of these conditions.

Another example that perhaps illuminates the significance of Minkowski’s theorem is the following. Consider a line through O having irrational slope tanθ; see Figure 4. This line passes through no lattice point other than the origin. If we take a long segment of this line, say extending length R on either side of O, then there will be a lattice point closest to, and a distance r from,

屏幕快照 2019-11-22 下午4.56.22.png

this segment. Hence, no matter how large R is, we can construct a rectangle containing this line segment, which contains no lattice point other than O. By the fundamental theorem of Minkowski the area 4rR of this rectangle does not exceed 4. Thus r1R. Note that if (p,q) is a lattice point on the border of the rectangle then pqtanθ, so that the fundamental theorem of Minkowski will give some information about how closely an irrational number can be approximated by rationals.

Let us now return to Hajos proof of the fundamental theorem of Minkowski. Consider the xy plane cut up into an infinite chessboard with the basic square of area 4 determined by |x|1, |y|1. We now cut up the chessboard along the edges of the squares and superimpose all the squares that contain parts of the region R. We have now compressed an area > 4 into a region of area 4. This implies that there will be some overlapping, i.e., one can stick a pin through the square so as to pierce R into two points say V1 and V2. Now reassemble the region and let the points V1 and V2 be the vectors V1 and V2. Consider the fact that the x and y coordinates of V1 and V2 differ by a multiple of 2. We write V1V2 (mod 2), which implies 12(V1V2)0 (mod 1). Thus 12(V1V2) is a lattice point different from O (since V1V2) in R.

The fundamental theorem of Minkowski can easily be generalized to n-dimensional space. Indeed we need only replace 4 in the fundamental theorem of Minkowski by 2n and Hajos’ proof goes through. Many extensions and re- finements of the fundamental theorem of Minkowski have been given. I shall return to some of them later.

One of Polya’s earliest papers has the long and curious title “Zahlhlentheoretisches und Wahrscheinlichkeitstheoretisches ¨uber die Sichtweite in Walde und durch Schneefall”. A proof of Polya’s main result in this paper can be greatly simplified and somewhat refined using the fundamental theorem of Minkowski. The problem is this.

Suppose every lattice point other than O is surrounded by a circle of radius r12 (a tree in a forest). A man stands at O. In direction θ he can see a distance f(r,θ). distance f(r,θ). What is the furthest he can see in any direction? That is, determine

F(r)=maxθf(θ,r)

屏幕快照 2019-11-22 下午5.05.10.png

By looking past the circle centered at (1, 0) (Figure 5), we can see almost a distance 1r. On the other hand we can prove that F(r)1r. For suppose that we can see a distance F(r) in direction θ. About this line of vision construct a rectangle with side 2r. This rectangle contains no lattice point, for otherwise the tree centered at such a lattice point would obstruct our line of vision; see Figure 6.

屏幕快照 2019-11-22 下午5.07.45.png

Hence, by the fundamental theorem of Minkowski 4F(r)r4 and F(r)1r as required. Note that no lattice point can be in either semicircle in the diagram. This enables us to improve slightly on Polya’s result. I shall leave the details as an exercise.

A more significant application of the fundamental theorem of Minkowski concerns the possibility of solving in integers a set of linear inequalities.

Consider the inequalities

|a11x1+a12x2++a1nxn|λ1,
|a21x1+a22x2++a2nxn|λ2,
.
.
.
|an1x1+an2x2++annxn|λn,

where the aij are real numbers and the λ1,λ2,...,λn are positive numbers. The problem is to find sufficient conditions for the existence of integers x1,...,xn, not all 0 satisfying the system. The fundamental theorem of Minkowski can be used to prove that a solution will exist provided the determinant det(aij) of the coefficients is, in absolute value, less than the product λ1λ2λn. This comes about in the following way. Geometrically, the inequalities determine an n−dimensional parallelepiped whose volume (or content) is

1det(aij)2nλ1λ2λn.

If λ1λ2λn>det(aij) then the content exceeds 2n and so contains a lattice point different from O.

A very recent analogue of the fundamental theorem of Minkowski is the following. Let R be a convex region, not necessarily symmetric about O, but having its centroid at O. If its area exceeds 92, then it contains a lattice point not O. The constant 92 is again best possible, but an n-dimensional analogue of this result is unknown.

The following is a conjectured generalization of the fundamental theorem of Minkowski, which we have unfortunately been unable to prove. Perhaps you will be able to prove or disprove it. Let R be a convex region containing the origin and defined by r=f(θ), 0θ<2π. If

π0f(θ)f(θ+π)dθ>4

then R contains a nontrivial lattice point. For symmetrical regions f(θ)=f(θ+π), and the conjecture reduces to the fundamental theorem of Minkowski.

Here is a somewhat related and only partially solved problem. Let M(n) be defined as the smallest number such that any convex region of area M(n) can be placed so as to cover n lattice points. Clearly M(1)=0. It is not difficult to show that M(2)=π4, i.e., any convex region whose area exceeds that of a circle of diameter 1 can be used to cover 2 lattice points. To determine M(3) already seems difficult. What one can easily prove is that M(n)n1 and we conjecture the existence of a positive constant c such that M(n)<ncn.


This page titled 1.8: Geometry of Numbers is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Leo Moser (The Trilla Group) via source content that was edited to the style and standards of the LibreTexts platform.

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