Gruber P. Convex and Discrete Geometry

26 The Torus Group E d /L 395
where β>0 is a suitable constant. Propositions (4) and (5), together with Blaschke’s
selection theorem 6.3, imply that a suitable subsequence of the sequence (Dn) converges
to a convex body D, where
ϱ
2 Bd ⊆ D ⊆ β B d .
It turns out that D is the Dirichlet–Voronoĭ cell of a lattice L. The subsequence of
(Ln) which corresponds to the convergent subsequence of the sequence of Dirichlet–
Voronoĭ cells, then converges to L. ⊓⊔
An Alternative Version
In some cases the following version of Mahler’s selection theorem is useful:
Corollary 25.1. The space L of lattices in E d is locally compact.
26 The Torus Group E d /L
A lattice is a sub-group of the additive group of E d . Given a lattice L, a natural
object to investigate is the quotient or torus group E d /L. Since this group is compact
and Abelian, there is a Haar measure m defined on it. In essence it is the ordinary
Lebesgue measure on a fundamental parallelotope. The question arises, to estimate
the measure
m(U + V)
of the sum U + V ={u + v : u ∈ U, v ∈ V} for measurable sets U, V in E d /L for
which U + V is also measurable. A satisfying answer to this question is the sum theorem
of Macbeath and Kneser. It was used by Kneser to prove a strong transference
theorem.
In this section, we first study the quotient group 〈E d /L, +〉, then present two
proofs of the sum theorem and, finally, use it to show Kneser’s transference theorem.
For information on topological groups, in particular for measure theory on topological
groups, see Nachbin [760].
26.1 Definitions and Simple Properties of E d /L
Let L be a lattice in E d . Then E d /L is a group which is endowed with a natural
topology and a natural measure. The topology on E d /L can be defined with a particular
notion of distance. E d /L thus carries a considerable lot of structure. Hence it is
plausible to expect interesting results.
In this section we define the group E d /L, a notion of distance which makes it a
topological group, and a measure. Both the distance and the measure are inherited
from E d in a simple way.