Radial Density in Apollonian Packings

TL;DR

This paper investigates the density of tangent circles in Apollonian packings near a given circle, using hyperbolic geometry and equidistribution, and extends the analysis to Soddy Sphere packings with explicit density limits.

Contribution

It establishes the limiting radial density in Apollonian packings using equidistribution on the modular surface and connects convergence rates to the Riemann Hypothesis, also analyzing Soddy Sphere packings.

Findings

Radial density in Apollonian packings is approximately 0.9549.

Convergence rate relates to the Riemann Hypothesis.

Soddy Sphere packings have a limiting density of about 0.853.

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 $\epsilon \rightarrow 0$? 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 $\pi/3$.