Radial Density in Apollonian Packings

Abstract

Given an Apollonian Circle Packing $\mathcal{P}$ and a circle $C_0=\partial B(z_0,r_0)$ in $\mathcal{P}$, color the set of disks in $\mathcal{P}$ tangent to $C_0$ red. What proportion of the concentric circle $C_{\epsilon }=\partial B(z_0,r_0+\epsilon )$ is red, and what is the behavior of this quantity as $\epsilon \to 0$? Using equidistribution of closed horocycles on the modular surface ${\mathbb{H}}^2/SL(2,\mathbb{Z})$, we show that the answer is $\frac{3}{\pi }=0.9549\dots$ 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 $\frac{\sqrt{3}}{2V_T}=0.853\dots$, where $V_T$ denotes the volume of an ideal hyperbolic tetrahedron with dihedral angles $\pi /3$.

New section based on an observation due to Alex Kontorovich connecting the rate of this convergence in the Farey-Ford packing to the Riemann Hypothesis