An Introduction to the Geometry of Numbers pp 121-174 | Cite as

# MAHLER’S compactness theorem

Chapter

## 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
are near those of the identity transformation, that is if

**M**being near to each other; and this is done by means of homogeneous linear transformations. A homogeneous linear transformation*=***X***of***τx***n*-dimensional euclidean space into itself is said to be near to identity transformation if the coefficients τ_{ ij }in$$ {X_i} = \sum\limits_{1\underline \le \,i\,\underline \le \,n} {{\tau _{ij}}{x_j}} \,\,\left( {1\,\underline \le \,i\,\underline \le \,n} \right) $$

$$ \left| {{\tau _{ii}} - \,1} \right|\,\,\,\left( {1\,\,\underline \le \,i\,\underline \le \,n} \right) $$

and
are all small.

$$ \left| {{\tau _{ij}}} \right|\,\,\,\,\left( {1\,\,\underline \le \,i\,\underline \le \,n,1\,\,\underline \le \,j\,\underline \le \,n,\,i \ne \,j\,} \right) $$

## 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