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Covering and packing in ${\Bbb Z}^n$ and ${\Bbb R}^n$, (I)

Authors:	Wolfgang M Schmidt David M Tuller

Institution:	(1) University of Colorado, Boulder, CO, USA

Abstract:	Given a finite subset ${\cal A}$ of an additive group ${\Bbb G}$ such as ${\Bbb Z}^n$ or ${\Bbb R}^n$ , we are interested in efficient covering of ${\Bbb G}$ by translates of ${\cal A}$ , and efficient packing of translates of ${\cal A}$ in ${\Bbb G}$ . A set ${\cal S} \subset {\Bbb G}$ provides a covering if the translates ${\cal A} + s$ with $s \in {\cal S}$ cover ${\Bbb G}$ (i.e., their union is ${\Bbb G}$ ), and the covering will be efficient if ${\cal S}$ has small density in ${\Bbb G}$ . On the other hand, a set ${\cal S} \subset {\Bbb G}$ will provide a packing if the translated sets ${\cal A} + s$ with $s \in {\cal S}$ are mutually disjoint, and the packing is efficient if ${\cal S}$ has large density. In the present part (I) we will derive some facts on these concepts when ${\Bbb G} = {\Bbb Z}^n$ , and give estimates for the minimal covering densities and maximal packing densities of finite sets ${\cal A} \subset {\Bbb Z}^n$ . In part (II) we will again deal with ${\Bbb G} = {\Bbb Z}^n$ , and study the behaviour of such densities under linear transformations. In part (III) we will turn to ${\Bbb G} = {\Bbb R}^n$ . Authors’ address: Department of Mathematics, University of Colorado at Boulder, Campus Box 395, Boulder, Colorado 80309-0395, USA The first author was partially supported by NSF DMS 0074531.

Keywords:	2000 Mathematics Subject Classification: 11H35 11B05 05B40
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