Corpus ID: 237091391

Asymptotic Density of Apollonian-Type Packings

@inproceedings{Litman2021AsymptoticDO,
  title={Asymptotic Density of Apollonian-Type Packings},
  author={M. Litman and A. Sheydvasser},
  year={2021}
}
There has been a lot of interest in studying “Apollonian-like” circle and sphere packings in recent years, with many different constructions and definitions [3, 5–7, 11, 19, 22]. Although there isn’t any unified agreement on what “Apollonian-like” should mean, at minimum it should be a set of oriented pn ́ 2q-spheres with non-intersecting interiors such that there exists a non-trivial subgroup G of O`pn, 1q acting on this set. For the most part, studying such sets has concentrated on taking… Expand

Figures and Tables from this paper

Is Unreasonable Slightness a General Phenomenon?
We consider the problem of determining when certain types of arithmetic groups like SLp2,Oq are generated by their elementary matrices. We give a simple, geometric criterion with a similarly simpleExpand

References

SHOWING 1-10 OF 28 REFERENCES
On Superintegral Kleinian Sphere Packings, Bugs, and Arithmetic Groups
We develop the notion of a Kleinian Sphere Packing, a generalization of “crystallographic” (Apollonian-like) sphere packings defined by Kontorovich-Nakamura [KN19]. Unlike crystallographic packings,Expand
Apollonian Circle Packings: Geometry and Group Theory II. Super-Apollonian Group and Integral Packings
TLDR
It is shown that the super-Apollonian group has finite volume in the group of all automorphisms of the parameter space of Descartes configurations, which is isomorphic to the Lorentz group O(3, 1). Expand
Geometry and arithmetic of crystallographic sphere packings
TLDR
This paper exhibits an infinite family of conformally inequivalent crystallographic packings with all radii being reciprocals of integers and proves a result in the opposite direction: the “superintegral” ones exist only in finitely many “commensurability classes,” all in, at most, 20 dimensions. Expand
Effective Bisector Estimate with Application to Apollonian Circle Packings
Let \Gamma<\PSL(2,\C) be a geometrically finite non-elementary discrete subgroup, and let its critical exponent \delta\ be greater than 1. We use representation theory of \PSL(2,\C) to prove anExpand
Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds
We obtain an asymptotic formula for the number of circles of curvature at most T in any given bounded Apollonian circle packing. For an integral packing, we obtain the upper bounds for the number ofExpand
A generalization of Apollonian packing of circles
TLDR
The construction provides a generalization of the Farey series and the associated Ford circles and the quantities bx/ √ 2 and by are integers. Expand
Effective Circle Count for Apollonian Packings and Closed Horospheres
The main result of this paper is an effective count for Apollonian circle packings that are either bounded or contain two parallel lines. We obtain this by proving an effective equidistribution ofExpand
Equidistribution and counting for orbits of geometrically finite hyperbolic groups
Let G be the identity component of SO(n,1), acting linearly on a finite dimensional real vector space V. Consider a vector w_0 in V such that the stabilizer of w_0 is a symmetric subgroup of G or theExpand
THE ASYMPTOTIC DISTRIBUTION OF CIRCLES IN THE ORBITS OF KLEINIAN GROUPS
Let P be a locally finite circle packing in the plane C invariant under a non-elementary Kleinian group Γ and with finitely many Γ-orbits. When Γ is geometrically finite, we construct an explicitExpand
The Unreasonable Slightness of E2 over Imaginary Quadratic Rings
  • Bogdan Nica
  • Mathematics, Computer Science
  • Am. Math. Mon.
  • 2011
TLDR
An elementary proof of this strong failure of elementary generation for SL2 over imaginary quadratic rings is given. Expand
...
1
2
3
...