A Theorem on Non-Homogeneous Lattices

Abstract

Tchebotareff, in his paper (11) dealing with Minkowski's conjecture, used a technique, depending on the additive properties of lattices, to show that, if the determinant d(A) is given, there is a point x of A such that a certain function f(x)-in his case I X1 X2 * * I -is less than a certain constant. The method depends on the construction of a sequence y', Y2, -*. , yr, ..* of points of A such that f(yr+i) < f(y,) and such that the sequence does not have to terminate until f is less than the prescribed bound. After the work of Chalk, using the same method on a different but related problem, it was natural to conjecture that the method would give a result about more general regions in Euclidean n-space (En). The generalization states, roughly, that, given a convex hypersurface R in En, and a positive number d; then there is another convex hypersurface R' contained in the convex domain bounded by R, having the following property. If A is any non-homogeneous lattice of determinant d, there is a point of A lying between R and R'. This generalization, which will be precisely stated in Section 2, is the subject of the present paper. The theorem has a number of interesting special cases, which are given at the end. The main difficulty of the proof lies not in constructing the sequence yi, Y2, ... , but in showing that f(yi) decreases quickly enough-that it does not, e.g., tend to a limit greater than the prescribed bound. This difficulty was easily overcome by Tchebotareff and Chalk, since they were able to use special properties of the particular function with which they were concerned. The difficulty is, however, quite formidable in the general case treated here, and its solution occupies most of the proof. Much use is made of the classical theory of convex sets, especially the Brunn-Minkowski Theorem, the Minkowski gaugefunction, and the selection theorem of Blaschke.