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By a Danzer set S we shall
mean a subset of the n-dimensional Euclidean space Rn which has the property that
every closed convex body of volume one in Rn 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 n-dimensional 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 max1≦i≦n|xi|≦ 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 →∞.
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