Vol. 37, No. 2, 1971

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On a problem of Danzer

Ram Prakash Bambah and Alan C. Woods

Vol. 37 (1971), No. 2, 295–301

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 max1in|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)n1 ) as X →∞.

Mathematical Subject Classification
Primary: 52A45
Received: 29 July 1970
Published: 1 May 1971
Ram Prakash Bambah
Alan C. Woods