Gruber P. Convex and Discrete Geometry

392 Geometry of Numbers
Topology on the Space of Lattices
We define the natural topology on the space L of all lattices in E d by specifying a
basis. A basis of the topology on L is given by the following sets of lattices:
� M ∈ L : M has a basis {c1,...,cd} with �ci − bi� 0.
The corresponding notion of convergence can be described as follows. A sequence
(Ln) of lattices in E d is convergent, if there is a lattice L in E d such that, for
suitable bases {bn1,...,bnd} of Ln, n = 1, 2,..., and {b1,...,bd} of L, respectively,
we have
bn1 → b1,...,bnd → bd as n →∞.
From now on, we assume that L is endowed with this topology.
It is not difficult to see that this topology is induced by a suitable metric on L.
See [447].
Dense Sets of Lattices
Confirming a statement of Rogers [851], Woods [1030] proved that the set of lattices,
where the basis vectors are the columns of the following matrices, is dense in the
space L(1):
⎛
α 0 ... 0 α1
⎜ 0 α ... 0 α2
⎜ ...............
⎝ 0 0 ... α αd−1
0 0 ... 0 α1−d ⎞
⎟
⎠ , where α1,...,αd−1 ∈ R and α>0.
This set of lattices can be used to prove the Minkowski–Hlawka theorem, see
Sect. 24.1.
A refinement of Wood’s result is the following result due to Schmidt [894]. For
almost all (d − 1)-tuples (β1,...,βd−1) ∈ Ed−1 , the set of lattices, where the basis
vectors are the columns of the matrices
⎛
⎞
is dense in L(1).
k 0 ... 0 β1k
⎜ 0 k ... 0 β2k ⎟
⎜ ................ ⎟ , k = 1, 2,...,
⎝
⎠
00... k βd−1k
00... 0 k 1−d
25.2 Mahler’s Selection Theorem
Mahler’s selection theorem is the main topological result for the space of lattices.
It provides a firm basis for several results which before were clear only intuitively.
In the following we present a standard version of it.