Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds

Kontorovich, Alex^1, 2 ; Oh, Hee

¹ Department of Mathematics, Brown University, Providence, Rhode Island 02912
² Department of Mathematics, Stony Brook University, Stony Brook, New York 11794-3651

Journal of the American Mathematical Society, Tome 24 (2011) no. 3, pp. 603-648

Résumé

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.

Détail

DOI : 10.1090/S0894-0347-2011-00691-7

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