By a Danzer set S we shall
mean a subset of the ndimensional Euclidean space R_{n} which has the property that
every closed convex body of volume one in R_{n} contains a point of S. L. Danzer has
asked if for n ≧ 2 there exist such sets S with a finite density. The answer to this
question is still unknown. In this note our object is to prove two theorems about
Danzer sets.
If Λ is a ndimensional lattice, any translate Γ = Λ + p of Λ will be called
a grid Γ;Λ will be called the lattice of Γ and the determinant d(Λ) of Λ
will be called the determinant of Γ and will be denoted by d(Γ). In §2 we
prove
Theorem 1. For n ≧ 2, a Danzer set cannot be the union of a finite number of
grids.
Let S be a Danzer sel and X > 0 a positive real number. Let N(S,X) be the
number of points of S in the box max_{1≦i≦n}x_{i}≦ X. Let D(S,X) = N(S,X)∕(2X)^{n}.
In §S we prove
Theorem 2. There exist Danzer sets S with D(S,X) = 0((log X)^{n−1
}) as
X →∞.
