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# MAHLER’S compactness theorem

• J. W. S. Cassels
Part of the Classics in Mathematics book series (volume 99)

## 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.

## Keywords

Convex Body Convergent Subsequence Compactness Theorem Diophantine Approximation Critical Lattice
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag Berlin Heidelberg 1997

## Authors and Affiliations

• J. W. S. Cassels
• 1
1. 1.Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeCambridgeUK