Given an Apollonian Circle Packing P and a circle C0β=βB(z0β,r0β) in P, color the set of disks in P tangent
to C0β red. What proportion of the concentric circle CΟ΅β=βB(z0β,r0β+Ο΅) is red, and what is the behavior of this quantity as
Ο΅β0? Using equidistribution of closed horocycles on the
modular surface H2/SL(2,Z), we show that the answer is
Ο3β=0.9549β¦ We also describe an observation due to Alex
Kontorovich connecting the rate of this convergence in the Farey-Ford packing
to the Riemann Hypothesis. For the analogous problem for Soddy Sphere packings,
we find that the limiting radial density is 2VTβ3ββ=0.853β¦,
where VTβ denotes the volume of an ideal hyperbolic tetrahedron with dihedral
angles Ο/3.Comment: New section based on an observation due to Alex Kontorovich
connecting the rate of this convergence in the Farey-Ford packing to the
Riemann Hypothesi