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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Apollonian circle packings and closed horospheres on hyperbolic $3$-manifolds
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by Alex Kontorovich and Hee Oh; with appendix by Hee Oh; with appendix by Nimish Shah PDF
J. Amer. Math. Soc. 24 (2011), 603-648 Request permission

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
  • MR Author ID: 704943
  • ORCID: 0000-0001-7626-8319
  • 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
  • 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.
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 24 (2011), 603-648
  • MSC (2010): Primary 22E40
  • DOI: https://doi.org/10.1090/S0894-0347-2011-00691-7
  • MathSciNet review: 2784325