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## Über dieses Buch

From the reviews: "The work is carefully written. It is well motivated, and interesting to read, even if it is not always easy... historical material is included... the author has written excellent account of an interesting subject." Mathematical Gazette "A well-written, very thorough account ... Among the topi are lattices, reduction, Minkowskis Theorem, distance functions, packings, and automorphs; some applications to number theory; excellent bibliographical references." The American Mathematical Monthly

## Inhaltsverzeichnis

### Prologue

Abstract
We owe to MINKOWSKI the fertile observation that certain results which can be made almost intuitive by the consideration of figures in n-dimensional euclidean space have far-reaching consequences in diverse branches of number theory. For example, he simplified the theory of units in algebraic number fields and both simplified and extended the theory of the approximation of irrational numbers by rational ones (Diophantine Approximation). This new branch of number theory, which MINKOWSKI christened “The Geometry of Numbers”, has developed into an independent branch of number-theory which, indeed, has many applications elsewhere but which is well worth studying for its own sake.
J. W. S. Cassels

### Chapter I. Lattices

Abstract
In this chapter we introduce the most important concept in the geometry of numbers, that of a lattice, and develop some of its basic properties. The contents of this chapter, except § 2.4 and § 5, are fundamental for almost everything that follows.
J. W. S. Cassels

### Chapter II. Reduction

Abstract
In investigating the values taken by an algebraic form fx) for integer values of the variables it is often useful to substitute for f a form equivalent to it (in the sense of Chapter I, § 4) which bears a special relation to the problem under consideration. This process is independent of the geometrical notions introduced by MINKOWSKI and depends only on the properties of bases of lattices developed in Chapter I. Indeed only the lattice Λ0 of integer vectors comes into consideration.
J. W. S. Cassels

### Chapter III. Theorems of BLICHFELDT and MINKOWSKI

Abstract
The whole of the geometry of numbers may be said to have sprung from MINKOWSKIS convex body theorem. In its crudest sense this says that if a point set L in n-dimensional euclidean space is symmetric about the origin (i.e. contains — x when it contains x) and convex [i.e. contains the whole line-segment λx + (1 – λ)y (0 ≦ λ ≦ 1)
when it contains x andy] and has volume V>2 n , then it contains an integral point u other than the origin. In this way we have a link between the “geometrical” properties of a set — convexity, symmetry and volume — and an “arithmetical” property, namely the existence of an integral point in L. Another form of the same theorem, which is more general only in appearance, states that if Λ is a lattice of determinant d(Λ) and L is convex and symmetric about the origin, as before, then L contains a point of Λ other than the origin, provided that the volume V of L is greater than 2 n d(Λ). In § 2 we shall prove MINKOWSKI’S theorem and some refinements. We shall not follow MINKOWSKI’S own proof but deduce his theorem from one of BLICHFELDT, which has important applications of its own and which is intuitively practically obvious: if a point set has volume strictly greater than d(Λ) then it contains two distinct points x1 and x 2 whose difference x 1 x 2 belongs to Λ.
J. W. S. Cassels

### Chapter IV. Distance-Functions

Abstract
In this chapter we introduce a number of concepts which are useful tools in all that follows.
J. W. S. Cassels

### Chapter V. MAHLER’S compactness theorem

Abstract
So far we have been concerned with one lattice at a time. In this chapter we are concerned with properties of sets of lattices. We first must define what is meant by two lattices Λ and M being near to each other; and this is done by means of homogeneous linear transformations. A homogeneous linear transformation X=τx of n-dimensional euclidean space into itself is said to be near to identity transformation if the coefficients τ ij in
$${X_i} = \sum\limits_{1\underline \le \,i\,\underline \le \,n} {{\tau _{ij}}{x_j}} \,\,\left( {1\,\underline \le \,i\,\underline \le \,n} \right)$$
are near those of the identity transformation, that is if
$$\left| {{\tau _{ii}} - \,1} \right|\,\,\,\left( {1\,\,\underline \le \,i\,\underline \le \,n} \right)$$
and
$$\left| {{\tau _{ij}}} \right|\,\,\,\,\left( {1\,\,\underline \le \,i\,\underline \le \,n,1\,\,\underline \le \,j\,\underline \le \,n,\,i \ne \,j\,} \right)$$
are all small.
J. W. S. Cassels

### Chapter VI. The theorem of MINKOWSKI-HLAWKA

Abstract
Hitherto we have been primarily concerned to estimate the lattice constant Δ (L) of a set L from below, that is to find numbers Δ 0 such that every lattice Λ with d (Λ) < Δ 0 certainly has points other than o inL In this chapter we are concerned with estimates for Δ(L) from above; that is we wish to find numbers Δ1 such that there are certainly lattices Λ with d(Λ) = Δ1 which have no points other than the origin in L, i.e. are L -admissible.
J. W. S. Cassels

### Chapter VII. The quotient space

Abstract
Before resuming the general study of the geometry of numbers, it is convenient to introduce here the concept of the quotient space of an n-dimensional space by a lattice. This concept plays an important rôle in the discussion of inhomogeneous problems in Chapter XI: but we shall also need it in Chapter VIII as it gives the most natural interpretation of Minkowski’S theorem about the successive minima of a convex body with respect to a lattice.
J. W. S. Cassels

### Chapter VIII. Successive minima

Abstract
For some purposes one requires to know not merely that a lattice Λ has a point in a set L, but that it has a number of linearly independent points in L.
J. W. S. Cassels

### Chapter IX. Packings

Abstract
If L is any n-dimensional set and y a point, we denote by L + y the set of points
L +y: x+y, xL. (1)
J. W. S. Cassels

### Chapter X. Automorphs

Abstract
A homogeneous linear transformation ω is said to be an automorph of a point set L if L is just the set of points ωx,xL. The automorphs of a set L evidently form a group. Many of the point sets of interest in the geometry of numbers, or which occur naturally in problems arising in other branches of number-theory, have a rich group of automorphs which is reflected in the set of L-admissible lattices. Already in the work in which he introduced the notion of limit of a sequence of lattices, MAHLER (1946d, e) laid the foundations for future work and indicated some fundamental theorems. Since then much has been done but some challenging and natural questions remain unanswered.
J. W. S. Cassels

### Chapter XI. Inhomogeneous problems

Abstract
As previously, we say that points x1 andx2 are congruent modulo Λ, written x1x2 (Λ),
where Λ is a lattice, to mean that x1x2∈Λ.
J. W. S. Cassels

### Backmatter

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