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Lattice packings with gap defects are not completely saturated
We show that a honeycomb circle packing in with a linear gap defect
cannot be completely saturated, no matter how narrow the gap is. The result is
motivated by an open problem of G. Fejes T\'oth, G. Kuperberg, and W.
Kuperberg, which asks whether of a honeycomb circle packing with a linear shift
defect is completely saturated. We also show that an fcc sphere packing in
with a planar gap defect is also not completely saturated
Notions of denseness
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.Comment: Published in Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol4/paper9.abs.htm
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