Packing and covering by translates of certain nonconvex bodies

We develop techniques for determining the packing and covering constants for star bodies composed of cubes. In the theory of convex sets problems of tiling, packing, and covering by translates of a given set have a long history, with the main focus on the packing and covering by spheres. Only in a few cases is the densest packing or sparsest covering known, even in the case of the sphere, except, of course, when the set happens to tile Eucidean space. In a series of papers S. K. Stein [4], [5], [6], [71 and W. Hamaker [1] used algebraic techniques in the problem of tiling Eucidean space of arbitrary dimension by translates of certain star bodies composed of cubes. The present paper has two purposes. First, it establishes a "Shift Theorem" that reduces tiling, packing, and covering problems for translates of a union of cubes to translates by vectors with integer coordinates. In a sense, this reduces continuous geometric problems in Eucidean space to discrete algebraic problems in a power of the infinite cyclic group. This theorem automatically generalizes many of Stein's results that depend on the assumption of integer translates. Second, it illustrates the Shift Theorem and a general "contribution" argument by determining the packing and covering constants for a particular star body. 1. Definitions. Consider in n-dimensional Euclidean space Rn the set of closed unit cubes parallel to the coordinate axes and whose centers have integer coordinates. The union of a finite number of these cubes, one of which has its center at the origin, is called a cluster, and denoted K. A family of translates of K whose union is Rn is a covering of Rn. A family of translates of K whose interiors are disjoint is a packing of R n. A covering that is also a packing is a tiling. Packing density and covering density of K are defined as they are for a convex body [3]. For S C Rn the interior of S will be denoted int(S). 2. The Shift Theorem. Let A be a subset of Rn homeomorphic to the ball En. Let A lattice-tile Rn, that is, assume there is a lattice L such that the family {I + A: / E L} tiles Rn. (For convenience, one may picture A as a unit square in the plane and the lattice L as the set of vectors with integer coordinates.) Let K be the union of some of the sets / + A; that is, let K = M + A for some M C L. We will show that the packing, covering, and tiling problems for K can be reduced to the case where the translation vectors are elements of L. Received by the editors August 10, 1978. AMS (MOS) subject classifications (1970). Primary 52A45, IOE30.