Gruber P. Convex and Discrete Geometry

408 Geometry of Numbers
Macbeath [674]. An easy proof led Macbeath to the following result: If L = AZ d ,
where A is a DOTU-matrix, then the conjecture holds for the lattice L. Itisthus
of interest to find out whether all d × d-matrices are DOTU-matrices or not. It is
easy to see that each real 2 × 2matrixisDOTU. After some unsuccessful attempts,
Narzulaev [763] established this for d = 3. Narzullaev and Ramharter [765] proved
that all non-singular 4 × 4 matrices A, for which the homogeneous minimum (2) of
the lattice L = AZ 4 is sufficiently small, are DOTU-matrices. It is easy to see that
for all d the rational d × d matrices are DOTU and Macbeath showed that the set
of DOTU-matrices is open. Hence, the d × dDOTU-matrices form a dense open
set in the space of all non-singular d × d matrices.
Unfortunately, Gruber [413] and Ahmedov [4] showed for infinitely many,
respectively, all sufficiently large d, the existence of d × d matrices which are not
DOTU. Skubenko [943] gave an example of such a matrix for d = 2880 and
Hendrik Lenstra [648] communicated an example with d = 64. These examples
make use of algebraic number theory and, in particular, of class field theory.
A Related Conjecture
which has attracted some interest is the following; we state it in geometric form:
Conjecture 27.3. For k = 0,...,d and any lattice L in Ed , the family of all translates
of the set
�
x : x1,...,xk ≥ 0, |x1 ···xd| ≤ d(L)
2d−k �
by vectors of L covers E d .
This conjecture was proved by Chalk [200] for k = d and all d, by Cole [212] for
k = d − 1 and all d and by Bambah and Woods [61] for k = 2 and d = 3. All other
cases seem to be open.
27.2 Mordell’s Inverse Problem and the Epstein Zeta-Function
A geometric version of the Minkowski linear form theorem is as follows, see
Corollary 22.2: Let L be a lattice in E d with d(L) = 1. Given numbers τ1,...,τd > 0
such that τ1 ···τd ≥ 1, the lattice L contains a point �= o in the box
(1) |x1| ≤τ1,...,|xd| ≤τd.
This led Mordell [753] to consider the problem to choose τ1,...,τd > 0 such that
τ1 ···τd is as large as possible and there is no point of L \{o} in the box (1).
Given a lattice L in E d ,theEpstein zeta-function ζ(L, ·) is defined as follows:
ζ(L, s) = �
l∈L\{o}
1 d
for s >
�l�2s 2 .