Volume 4, issue 1 (2000)

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Notions of denseness

Greg Kuperberg

Geometry & Topology 4 (2000) 277–292
 arXiv: math.MG/9908003
Abstract

The notion of a completely saturated packing [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
Mathematical Subject Classification 2000
Primary: 52C15, 52C17
Secondary: 52C20, 52C22, 52C26, 52B99