Gruber P. Convex and Discrete Geometry

396 Geometry of Numbers
The Group E d /L
With respect to vector addition +, the space E d is an Abelian group and L a subgroup
of it. The torus or quotient group E d /L is the Abelian group consisting of all
cosets of L, that is translates L + x ={l + x : l ∈ L} of L, where x ∈ E d .ForL + x
we also write x. Addition + on E d /L is defined by:
x + y = L + x + L + y = L + x + y for x = L + x, y = L + y ∈ E d /L.
Clearly, addition on E d /L is independent of the particular choice of x, y. A coset
x = L + x is also called an inhomogeneous lattice. Afundamental domain of E d /L
is a subset of E d which contains precisely one point of each coset. If {b1,...,bd} is
a basis of L, then the corresponding fundamental parallelotope
F = � α1b1 +···+αdbd : 0 ≤ αi < 1 �
is a fundamental domain. Addition in E d /L corresponds to addition modulo L in F,
symbolized by + L (Fig. 26.1):
x + L y = (L + x + y) ∩ F for x, y ∈ F.
Proposition 26.1. The groups 〈E d /L, +〉 and 〈F, + L 〉 are isomorphic. An isomorphism
is given by the mapping
x → x ∩ F for x ∈ E d /L.
Proof. Left to the reader. ⊓⊔
Topology on E d /L
Define a distance modulo L, �·� L ,onE d /L by:
�x� L = inf � �l + x� :l ∈ L � for x = L + x ∈ E d /L.
Since L is discrete, x = L + x is also discrete. Hence the infimum is attained for a
suitable l ∈ L. The distance �·� L yields a metric and thus induces a topology on
E d /L. We assume, from now on, that E d /L is endowed with this topology.
o
y
x
x + L y
x + y
Fig. 26.1. Addition modulo L