Gruber P. Convex and Discrete Geometry

490 Geometry of Numbers
For the proof of (20) let λ>1. Since J is Jordan measurable, the following statement
holds:
(22) There are cubes C1,...,Ck such that:
J ⊆ C1 ∪···∪Ck, Ci ∩ J �= ∅for i = 1,...,k,
λV (J)
V (Ci) all are equal and ≤ ,
k
�
V (Ci) α+d
d ≤ (λ − 1) �
V (Ci) α+d
d .
Ci �⊆J
Using this, we prove that
(23) I (J,α,n) < ∼ λ 3+ α d δV (J) α+d
d
1
n α d
as n →∞.
At first, the case where n = kl, l = 1, 2,...,will be considered. For i = 1,...,k
and l = 1, 2,..., choose minimizing configurations Sil for I (C ∩ J,α,l) with
#Sil = l. LetSn = �
Sil. Then #Sn ≤ n = kl. The definition of I together with
(22), (21), (11) and (22) shows that
i
I (J,α,n = kl) n α �
d ≤ min{�x
− s�
s∈Sn
J
α } dx n α d
≤ �
�
min{�x
− s�
s∈Sil
i
Ci ∩J
α } dx n α d
� �
≤ I (Ci,α,l) + �
�
I (Ci ∩ J,α,l) n α d
� �
≤ I (Ci,α,l) + �
Ci ⊆J
� �
∼ δ
Ci ⊆J
Ci ⊆J
i
i
Ci �⊆J
Ci �⊆J
V (Ci) α+d
d + �
Ci �⊆J
�
I (Ci,α,l) n α d
V (Ci) α+d �
d k α d
� �
≤ δ V (Ci) α+d
d + (λ − 1) �
V (Ci) α+d �
d k α d
≤ λδ �
i
V (Ci) α+d
d k α d = λδk
= λ 2+ α d δV (J) α+d
d as l →∞.
i
� λV (J)
k
� α+d
d k α d
Thus (23) holds for n of the form n = kl, l = 1, 2,... with λ 2+α/d instead of
λ 3+α/d . A similar argument as the one that led to (10), then yields (23).
Since λ>1 was arbitrary, (23) implies (20). The desired asymptotic formula for
I (J,α,n) finally follows from (12) and (20). ⊓⊔