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


Authors: Alex Kontorovich and Hee Oh; with appendix by Hee Oh; Nimish Shah
Journal: J. Amer. Math. Soc. 24 (2011), 603-648
MSC (2010): Primary 22E40
DOI: https://doi.org/10.1090/S0894-0347-2011-00691-7
Published electronically: January 19, 2011
MathSciNet review: 2784325
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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.


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Additional Information

Alex Kontorovich
Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912
Address at time of publication: Department of Mathematics, Stony Brook University, Stony Brook, New York 11794-3651
Email: alexk@math.brown.edu, alexk@math.sunysb.edu

Hee Oh
Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912 and Department of Mathematics, Korea Institute for Advanced Study, Seoul, Korea
Email: heeoh@math.brown.edu

Nimish Shah
Affiliation: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210
Email: shah@math.ohio-state.edu

DOI: https://doi.org/10.1090/S0894-0347-2011-00691-7
Keywords: Apollonian circle packing, Horospheres, Kleinian group
Received by editor(s): January 20, 2009
Received by editor(s) in revised form: November 18, 2009, and December 10, 2010
Published electronically: January 19, 2011
Additional Notes: The first author is supported by an NSF Postdoc, grant DMS 0802998.
The second author is partially supported by NSF grant DMS 0629322.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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