Notions of denseness

Greg Kuperberg

Notions of denseness

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arxiv math/9908003 v3 pith:2VG5SVXE submitted 1999-08-01 math.MG

Greg Kuperberg This is my paper · ORCID

classification math.MG

keywords completelypackingpackingsrecurrentsaturatedsharperdensedensity

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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.

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Variations on five-dimensional sphere packings
math.MG 2024-12 accept novelty 7.0 of 10

New non-isometric kissing configurations are constructed in dimensions 5 and 9, and the uniform 5D packings containing the known kissing configurations are classified.

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