Radial density in Apollonian packings

Given P, an Apollonian Circle Packing, and a circle C₀ =∂ B(z₀, r₀) in P, color the set of disks in P tangent to C₀ red. What proportion of the concentric circle Cε =∂ B(z₀, r₀ + ε) is red, and what is the behavior of this quantity as ε→0? Using equidistribution of closed horocycles on the modular surface ℍ²/SL(2,ℤ), we show that the answer is 3/π =0.9549 . 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 √3/2V_T =0.853 ., where V_T denotes the volume of an ideal hyperbolic tetrahedron with dihedral angles π/3.