Geometry & Topology, Vol. 4 (2000) Paper no. 9, pages 277--292.

Notions of denseness

Greg Kuperberg

Abstract. The notion of a completely saturated packing [Fejes Toth, Kuperberg and Kuperberg, Highly saturated packings and reduced coverings, Monats. Math. 125 (1998) 127-145] is a sharper version of maximum density, and the analogous notion of a completely reduced covering is a sharper version of minimum density. We define two related notions: uniformly recurrent and weakly recurrent dense packings, and diffusively dominant packings. Every compact domain in Euclidean space has a uniformly recurrent dense packing. If the domain self-nests, such a packing is limit-equivalent to a completely saturated one. Diffusive dominance is yet sharper than complete saturation and leads to a better understanding of n-saturation.

Keywords. Density, saturation, packing, covering, dominance

AMS subject classification. Primary: 52C15, 52C17. Secondary: 52C20, 52C22, 52C26, 52B99.

DOI: 10.2140/gt.2000.4.277

E-print: arXiv:math.MG/9908003

Submitted to GT on 4 August 1999. (Revised 28 September 2000.) Paper accepted 21 September 2000. Paper published 8 October 2000.

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Greg Kuperberg
Department of Mathematics, University of California
One Shields Ave, Davis, CA 95616-8633, USA

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