Gruber P. Convex and Discrete Geometry

368 Geometry of Numbers
we associate the point
w1
n b1 +···+ wd
n bd where wi ≡ ui mod n and wi ∈{0,...,n − 1}.
There are precisely nd points of the latter form. Since, by (2), there are more than
nd points in 1 2C ∩ 1 n L, the Dirichlet pigeon hole principle implies that there are two
distinct points, say
u1
n b1 +···+ ud
n
bd, v1
n b1 +···+ vd
n bd ∈ 1 1
C ∩
2 n L,
for which the associated points coincide. Then ui ≡ vi mod n, orn|(ui − vi) for
i = 1,...,d, and we obtain
o �= u1 − v1
n
� �� �
∈Z
b1 +···+ ud − vd
n
� �� �
∈Z
bd ∈
�
1 1
C −
2 2 C
�
∩ L = C ∩ L.
This concludes the first proof of the fundamental theorem. ⊓⊔
Proof (of Siegel with Fourier series). It is sufficient to show the following proposition:
(3) Let C ∩ L ={o}. Then V (C) ≤ 2 d d(L).
For the proof of (3) we assume that C ∩ L ={o} and first show the following:
(4) The convex bodies 1 2C + l, l ∈ L, are pairwise disjoint.
If (4) did not hold, then 1 2 x + l = 1 2 y + m for suitable x, y ∈ C and l, m ∈ L,
l �= m. Hence o �= l − m = 1 2 y − 1 2 x ∈ ( 1 2C − 1 2C) ∩ L = C ∩ L. This contradicts
the assumption in (3) and thus concludes the proof of (4).
Let 1 be the characteristic function of 1 2C. Clearly, the function ψ : Ed → R
defined by:
(5) ψ(x) = �
1(x + l) is L-periodic.
l∈L
Because of (4), we have ψ(x) = 0 or 1 for each x ∈ E d and thus
(6) ψ 2 = ψ.
To ψ corresponds the Fourier series
�
c(m)e 2πi m·x ,
m∈L ∗
where, for the Fourier coefficients c(m), we have the following representations: