Gruber P. Convex and Discrete Geometry

Asymptotic Estimates
27 Special Problems in the Geometry of Numbers 407
Chebotarev [202] proved that, for each lattice L, the family of all translates of the set
|x1 ···xd| ≤ d(L)
by vectors of L covers E d . The proof uses a clever application of Minkowski’s first
fundamental theorem. Using refinements of Minkowski’s theorem, many successive
improvements of this result were given, where 2 d/2 is replaced by νd2 d/2 with
νd > 1. We mention the following; for others, see the author and Lekkerkerker [447]
and Bambah, Dumir and Hans-Gill [58]:
νd ∼ 2e − 1: Davenport [244]
νd ∼ 2(2e − 1): Woods [1029]
νd ∼ 3.0001(2e − 1): Bombieri [147]
νd ∼ 3(2e − 1): Gruber [410]
νd ∼ e − 2 3 d 1 3 log − 2 3 d as d →∞Skubenko [942]
See also Narzullaev and Skubenko [764], Mukhsinov [759] and Andriyasyan, Il’in
and Malyshev [34].
Sets of Lattices for which the Conjecture is true and DOTU-Matrices
A result of Birch and Swinnerton-Dyer [117] is as follows: If the conjecture holds
for dimensions 2,...,d − 1, then it holds in dimension d for all lattices L for which
the homogeneous minimum
(2) λ(L) = inf � |l1 ···ld| : l ∈ L \{o} �
is 0. Since it is easy to prove that λ(L) = 0 for almost all lattices L ∈ L(1) in the
sense of the measure introduced by Siegel on the space of all lattices of determinant
1, we see that, if the conjecture holds in dimensions 2,...,d − 1, then it holds in
dimension d for almost all lattices of determinant 1, see [411]. In the same article,
it was shown that, in all dimensions, the measure of the set of lattices of determinant
1 which cover Ed by the set
|x1 ···xd| ≤ dd!
√
2π
∼ √
dd ded = e−d+o(d) as d →∞,
is at least 1 − e −0.279 d .
A real d × d-matrix A is a DOTU-matrix if it can be written in the form:
2 d 2
A = DOTU,
where D is a diagonal, O an orthogonal, T an upper triangular matrix with 1’s in the
diagonal and U an integer unimodular d × d matrix. This notion was introduced by