Gruber P. Convex and Discrete Geometry

23 Successive Minima 383
as required. This representation of P ∗ immediately yields the inclusions
which, in turn, show that
(3) λQ ≤ λ1(P ∗ , Z d ) ≤ d λQ.
1
d Q ⊆ P∗ ⊆ Q
The theorem on successive minima implies the following:
λ d−1
P
≤ λ1(P, Z d ) ···λd−1(P, Z d ) ≤
2d λd(P, Zd | det A|
=
)V (P) λd(P, Zd ) .
This, together with Theorem 23.2 and Proposition (3), shows that
λ d−1
P
≤ λ1(P ∗ , Z d )| det A| ≤d λQ| det A|.
This yields the first inequality. The second follows by symmetry. ⊓⊔
The following result goes back to Perron [793] and Khintchine [581]. It shows
that the results on simultaneous Diophantine approximation and on approximation
of linear forms in Sect. 22.2 are closely related.
Theorem 23.5. Let ϑ1,...,ϑd ∈ R. Then the following propositions are equivalent:
(i) There is a constant α>0 such that the following system of inequalities has no
integer solution (u0,...,ud) where u0 �= 0:
�
�
�ϑ1 − u1
�
�
� ≤ α
�
�
,..., �ϑd − ud
�
�
� ≤ α
.
u0
u 1+ 1 d
0
u0
u 1+ 1 d
0
(ii) There is a constant β > 0 such that the following inequality has no integer
solution (u0,...,ud) where (u1,...,ud) �= o:
|u1ϑ1 +···+udϑd − u0| ≤
β
.
max{|u1|,...,|ud|} d
The following interpretation may help in understanding the meaning of this result:
ϑ1,...,ϑd cannot be simultaneously approximated well by rationals with the same
denominator if and only if the linear form u1ϑ1 +···+udϑd, for integers u1,...,ud
not all 0, cannot be approximated well by integers.
Proof. (i)⇒(ii) If (i) holds, then, for each τ>α d , the following system of inequalities
has only the trivial integer solution:
|u0ϑ1 − u1| ≤ α
,...,|u0ϑd − ud| ≤ α
, |u0| ≤τ.
τ 1 d
τ 1 d