Affiliation:
1. Department of Mathematics , UC Davis , Davis , CA 95616 , USA
2. Rutgers University , New Brunswick , NJ 08901-8554; and National Museum of Mathematics, NY , USA
Abstract
Abstract
We develop the notion of a Kleinian Sphere Packing, a generalization of
“crystallographic” (Apollonian-like) sphere packings defined in [A. Kontorovich and K. Nakamura,
Geometry and arithmetic of crystallographic sphere packings,
Proc. Natl. Acad. Sci. USA 116 2019, 2, 436–441].
Unlike crystallographic packings, Kleinian packings exist in all dimensions, as do “superintegral” such.
We extend the Arithmeticity Theorem to Kleinian packings, that is, the superintegral ones come from
ℚ
{{\mathbb{Q}}}
-arithmetic lattices of simplest type.
The same holds for more general objects we call Kleinian Bugs, in which the spheres need not be disjoint but can meet with dihedral angles
π
m
{\frac{\pi}{m}}
for finitely many m.
We settle two questions from Kontorovich and Nakamura (2019): (i) that the Arithmeticity Theorem is in general false over number fields, and (ii)
that integral packings only arise from non-uniform lattices.
Subject
Applied Mathematics,General Mathematics
Cited by
2 articles.
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