Apollonian circle packings and closed horospheres on hyperbolic $3$-manifolds

Abstract:

We show that for a given bounded Apollonian circle packing $\mathcal P$, there exists a constant $c>0$ such that the number of circles of curvature at most $T$ is asymptotic to $c\cdot T^\alpha$ as $T\to \infty$. Here $\alpha \approx 1.30568(8)$ is the residual dimension of the packing. For $\mathcal P$ integral, let $\pi ^{\mathcal {P}}(T)$ denote the number of circles with prime curvature less than $T$. Similarly let $\pi _2^{\mathcal {P}}(T)$ be the number of pairs of tangent circles with prime curvatures less than $T$. We obtain the upper bounds $\pi ^{\mathcal {P}}(T)\ll T^\alpha /\log T$ and $\pi _2^{\mathcal {P}}(T)\ll T^\alpha /(\log T)^2$, which are sharp up to a constant multiple. The main ingredient of our proof is the effective equidistribution of expanding closed horospheres in the unit tangent bundle of a geometrically finite hyperbolic $3$-manifold $\Gamma \backslash \mathbb {H}^3$ under the assumption that the critical exponent of $\Gamma$ exceeds one.