Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds

TL;DR

This paper derives an asymptotic count for circles in Apollonian packings, including prime curvature bounds, using horosphere equidistribution on hyperbolic 3-manifolds.

Contribution

It provides the first effective asymptotic formulas for circle counts and prime curvature bounds in Apollonian packings via hyperbolic geometry methods.

Findings

Asymptotic formula for circle counts in Apollonian packings

Upper bounds for prime curvature circles and pairs

Effective equidistribution results on hyperbolic 3-manifolds

Abstract

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 of circles with prime curvature as well as of pairs of circles with prime curvatures, which are sharp up constant multiples. The main ingredient of our proof is the effective equidistribution of expanding horospheres on geometrically finite hyperbolic 3-manifolds under the assumption that the critical exponent of its fundamental group exceeds one.