## 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
- J. Amer. Math. Soc.
**24**(2011), 603-648 - DOI: https://doi.org/10.1090/S0894-0347-2011-00691-7
- Published electronically: January 19, 2011
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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.## References

- Thierry Aubin,
*Nonlinear analysis on manifolds. Monge-Ampère equations*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 252, Springer-Verlag, New York, 1982. MR**681859**, DOI 10.1007/978-1-4612-5734-9 - Alan F. Beardon and Bernard Maskit,
*Limit points of Kleinian groups and finite sided fundamental polyhedra*, Acta Math.**132**(1974), 1–12. MR**333164**, DOI 10.1007/BF02392106 - A. I. Borevich and I. R. Shafarevich,
*Number theory*, Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. Translated from the Russian by Newcomb Greenleaf. MR**0195803** - Jean Bourgain, Alex Gamburd, and Peter Sarnak,
*Affine linear sieve, expanders, and sum-product*, Invent. Math.**179**(2010), no. 3, 559–644. MR**2587341**, DOI 10.1007/s00222-009-0225-3 - Jean Bourgain, Alex Gamburd, and Peter Sarnak. Generalization of Selberg’s theorem and Selberg’s sieve, 2009. Preprint.
- David W. Boyd,
*The sequence of radii of the Apollonian packing*, Math. Comp.**39**(1982), no. 159, 249–254. MR**658230**, DOI 10.1090/S0025-5718-1982-0658230-7 - Marc Burger,
*Horocycle flow on geometrically finite surfaces*, Duke Math. J.**61**(1990), no. 3, 779–803. MR**1084459**, DOI 10.1215/S0012-7094-90-06129-0 - Kevin Corlette and Alessandra Iozzi,
*Limit sets of discrete groups of isometries of exotic hyperbolic spaces*, Trans. Amer. Math. Soc.**351**(1999), no. 4, 1507–1530. MR**1458321**, DOI 10.1090/S0002-9947-99-02113-3 - H. S. M. Coxeter,
*The problem of Apollonius*, Amer. Math. Monthly**75**(1968), 5–15. MR**230204**, DOI 10.2307/2315097 - Françoise Dal’bo,
*Topologie du feuilletage fortement stable*, Ann. Inst. Fourier (Grenoble)**50**(2000), no. 3, 981–993 (French, with English and French summaries). MR**1779902**, DOI 10.5802/aif.1781 - S. G. Dani,
*Invariant measures and minimal sets of horospherical flows*, Invent. Math.**64**(1981), no. 2, 357–385. MR**629475**, DOI 10.1007/BF01389173 - Alex Eskin and Curt McMullen,
*Mixing, counting, and equidistribution in Lie groups*, Duke Math. J.**71**(1993), no. 1, 181–209. MR**1230290**, DOI 10.1215/S0012-7094-93-07108-6 - E. Fuchs. Ph.D. Thesis, Princeton University, 2010.
- Ramesh Gangolli and V. S. Varadarajan,
*Harmonic analysis of spherical functions on real reductive groups*, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 101, Springer-Verlag, Berlin, 1988. MR**954385**, DOI 10.1007/978-3-642-72956-0 - Ronald L. Graham, Jeffrey C. Lagarias, Colin L. Mallows, Allan R. Wilks, and Catherine H. Yan,
*Apollonian circle packings: number theory*, J. Number Theory**100**(2003), no. 1, 1–45. MR**1971245**, DOI 10.1016/S0022-314X(03)00015-5 - Ronald L. Graham, Jeffrey C. Lagarias, Colin L. Mallows, Allan R. Wilks, and Catherine H. Yan,
*Apollonian circle packings: geometry and group theory. I. The Apollonian group*, Discrete Comput. Geom.**34**(2005), no. 4, 547–585. MR**2173929**, DOI 10.1007/s00454-005-1196-9 - Y. Guivarc’h and A. Raugi,
*Products of random matrices: convergence theorems*, Random matrices and their applications (Brunswick, Maine, 1984) Contemp. Math., vol. 50, Amer. Math. Soc., Providence, RI, 1986, pp. 31–54. MR**841080**, DOI 10.1090/conm/050/841080 - K. E. Hirst,
*The Apollonian packing of circles*, J. London Math. Soc.**42**(1967), 281–291. MR**209981**, DOI 10.1112/jlms/s1-42.1.281 - Henryk Iwaniec and Emmanuel Kowalski,
*Analytic number theory*, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR**2061214**, DOI 10.1090/coll/053 - Michael Kapovich,
*Hyperbolic manifolds and discrete groups*, Progress in Mathematics, vol. 183, Birkhäuser Boston, Inc., Boston, MA, 2001. MR**1792613** - Alex V. Kontorovich,
*The hyperbolic lattice point count in infinite volume with applications to sieves*, Duke Math. J.**149**(2009), no. 1, 1–36. MR**2541126**, DOI 10.1215/00127094-2009-035 - Alex Kontorovich and Hee Oh. Almost prime Pythagorean triples in thin orbits.
*Preprint, 2009*, arXiv:1001.0370. - Serge Lang,
*Algebra*, 3rd ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR**1878556**, DOI 10.1007/978-1-4613-0041-0 - Peter D. Lax and Ralph S. Phillips,
*The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces*, J. Functional Analysis**46**(1982), no. 3, 280–350. MR**661875**, DOI 10.1016/0022-1236(82)90050-7 - Grigoriy A. Margulis,
*On some aspects of the theory of Anosov systems*, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004. With a survey by Richard Sharp: Periodic orbits of hyperbolic flows; Translated from the Russian by Valentina Vladimirovna Szulikowska. MR**2035655**, DOI 10.1007/978-3-662-09070-1 - Bernard Maskit,
*Kleinian groups*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. MR**959135** - C. R. Matthews, L. N. Vaserstein, and B. Weisfeiler,
*Congruence properties of Zariski-dense subgroups. I*, Proc. London Math. Soc. (3)**48**(1984), no. 3, 514–532. MR**735226**, DOI 10.1112/plms/s3-48.3.514 - François Maucourant,
*Homogeneous asymptotic limits of Haar measures of semisimple linear groups and their lattices*, Duke Math. J.**136**(2007), no. 2, 357–399. MR**2286635**, DOI 10.1215/S0012-7094-07-13626-3 - Curtis T. McMullen,
*Hausdorff dimension and conformal dynamics. III. Computation of dimension*, Amer. J. Math.**120**(1998), no. 4, 691–721. MR**1637951**, DOI 10.1353/ajm.1998.0031 - Amir Mohammadi and Alireza Salehi Golsefidy. Translates of horospherical measures and counting problems.
*Preprint*, 2008. - Hee Oh. Dynamics on Geometrically finite hyperbolic manifolds with applications to Apollonian circle packings and beyond.
*Proc. of ICM (Hyderabad, 2010)*. - Hee Oh and Nimish Shah. Equidistribution and counting for orbits of geometrically finite hyperbolic groups.
*Preprint*, arXiv:1001.2096. - Hee Oh and Nimish Shah. The asymptotic distribution of circles in the orbits of Kleinian groups.
*Preprint*, arXiv:1004.2130. - S. J. Patterson,
*The limit set of a Fuchsian group*, Acta Math.**136**(1976), no. 3-4, 241–273. MR**450547**, DOI 10.1007/BF02392046 - Vladimir Platonov and Andrei Rapinchuk,
*Algebraic groups and number theory*, Pure and Applied Mathematics, vol. 139, Academic Press, Inc., Boston, MA, 1994. Translated from the 1991 Russian original by Rachel Rowen. MR**1278263** - Marina Ratner,
*On Raghunathan’s measure conjecture*, Ann. of Math. (2)**134**(1991), no. 3, 545–607. MR**1135878**, DOI 10.2307/2944357 - Marina Ratner,
*Raghunathan’s topological conjecture and distributions of unipotent flows*, Duke Math. J.**63**(1991), no. 1, 235–280. MR**1106945**, DOI 10.1215/S0012-7094-91-06311-8 - Thomas Roblin,
*Ergodicité et équidistribution en courbure négative*, Mém. Soc. Math. Fr. (N.S.)**95**(2003), vi+96 (French, with English and French summaries). MR**2057305**, DOI 10.24033/msmf.408 - Peter Sarnak,
*Asymptotic behavior of periodic orbits of the horocycle flow and Eisenstein series*, Comm. Pure Appl. Math.**34**(1981), no. 6, 719–739. MR**634284**, DOI 10.1002/cpa.3160340602 - Peter Sarnak. Letter to J. Lagarias, 2007. available at www.math.princeton.edu/$\sim$sarnak.
- Barbara Schapira,
*Equidistribution of the horocycles of a geometrically finite surface*, Int. Math. Res. Not.**40**(2005), 2447–2471. MR**2180113**, DOI 10.1155/IMRN.2005.2447 - Yehuda Shalom,
*Rigidity, unitary representations of semisimple groups, and fundamental groups of manifolds with rank one transformation group*, Ann. of Math. (2)**152**(2000), no. 1, 113–182. MR**1792293**, DOI 10.2307/2661380 - F. Soddy. The bowl of integers and the hexlet.
*Nature*, 139:77–79, 1936. - F. Soddy. The kiss precise.
*Nature*, 137:1021, 1937. - Dennis Sullivan,
*The density at infinity of a discrete group of hyperbolic motions*, Inst. Hautes Études Sci. Publ. Math.**50**(1979), 171–202. MR**556586**, DOI 10.1007/BF02684773 - Dennis Sullivan,
*Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups*, Acta Math.**153**(1984), no. 3-4, 259–277. MR**766265**, DOI 10.1007/BF02392379 - William Thurston.
*The Geometry and Topology of Three-Manifolds*. available at www.msri.org/publications/books. Electronic version-March 2002. - J. B. Wilker,
*Sizing up a solid packing*, Period. Math. Hungar.**8**(1977), no. 2, 117–134. MR**641055**, DOI 10.1007/BF02018498

## Bibliographic 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