## A proof of the positive density conjecture for integer Apollonian circle packings

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- by Jean Bourgain and Elena Fuchs
- J. Amer. Math. Soc.
**24**(2011), 945-967 - DOI: https://doi.org/10.1090/S0894-0347-2011-00707-8
- Published electronically: June 20, 2011
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## Abstract:

An Apollonian circle packing (ACP) is an ancient Greek construction which is made by repeatedly inscribing circles into the triangular interstices in a Descartes configuration of four mutually tangent circles. Remarkably, if the original four circles have integer curvature, all of the circles in the packing will have integer curvature as well. In this paper, we compute a lower bound for the number $\kappa (P,X)$ of integers less than $X$ occurring as curvatures in a bounded integer ACP $P$, and prove a conjecture of Graham, Lagarias, Mallows, Wilkes, and Yan that the ratio $\kappa (P,X)/X$ is greater than $0$ for $X$ tending to infinity.## References

- P. Bernays,
*Über die Darstellung von positiven, ganzen Zahlen durch die primitiven, binären quadratischen Formen einer nicht quadratischen Diskriminante*, Ph.D. dissertation, Georg-August-Universität, Göttingen, Germany (1912). - Valentin Blomer and Andrew Granville,
*Estimates for representation numbers of quadratic forms*, Duke Math. J.**135**(2006), no. 2, 261–302. MR**2267284**, DOI 10.1215/S0012-7094-06-13522-6 - 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 - J. W. S. Cassels,
*Rational quadratic forms*, London Mathematical Society Monographs, vol. 13, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1978. MR**522835** - H. S. M. Coxeter,
*An absolute property of four mutually tangent circles*, Non-Euclidean geometries, Math. Appl. (N. Y.), vol. 581, Springer, New York, 2006, pp. 109–114. MR**2191243**, DOI 10.1007/0-387-29555-0_{5} - W. Duke, Z. Rudnick, and P. Sarnak,
*Density of integer points on affine homogeneous varieties*, Duke Math. J.**71**(1993), no. 1, 143–179. MR**1230289**, DOI 10.1215/S0012-7094-93-07107-4 - J. Elstrodt, F. Grunewald, and J. Mennicke,
*Groups acting on hyperbolic space*, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. Harmonic analysis and number theory. MR**1483315**, DOI 10.1007/978-3-662-03626-6 - T. Estermann,
*A new application of the Hardy-Littlewood-Kloosterman method*, Proc. London Math. Soc. (3)**12**(1962), 425–444. MR**137677**, DOI 10.1112/plms/s3-12.1.425 - John Friedlander and Henryk Iwaniec,
*Opera de cribro*, American Mathematical Society Colloquium Publications, vol. 57, American Mathematical Society, Providence, RI, 2010. MR**2647984**, DOI 10.1090/coll/057 - E. Fuchs,
*Arithmetic properties of Apollonian circle packings*, Ph.D. Thesis, Princeton (2010). - E. Fuchs,
*A note on the density of curvatures in integer Apollonian circle packings*, preprint, http://www.math.ias.edu/~efuchs (2009). - E. Fuchs, K. Sanden,
*Some experiments with integral Apollonian circle packings*, J. Exp. Math., to appear. - 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 - D. R. Heath-Brown,
*A new form of the circle method, and its application to quadratic forms*, J. Reine Angew. Math.**481**(1996), 149–206. MR**1421949**, DOI 10.1515/crll.1996.481.149 - Edward Kasner and Fred Supnick,
*The Apollonian packing of circles*, Proc. Nat. Acad. Sci. U.S.A.**29**(1943), 378–384. MR**9128**, DOI 10.1073/pnas.29.11.378 - 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 - Svetlana Katok,
*Fuchsian groups*, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992. MR**1177168** - H.D. Kloosterman,
*On the representation of numbers of the form $ax^2+by^2+cz^2+dt^2$*, Acta Math.**49**, pp. 407-464 (1926). - A. Kontorovich, H. Oh,
*Apollonian circle packings and closed horospheres on hyperbolic $3$-manifolds*, J. Amer. Math. Soc.**24**, pp. 603–648 (2011). - N. Niedermowwe,
*A version of the circle method for the representation of integers by quadratic forms*, preprint arXiv:0905.1229v1 (2009). - K. Sanden,
*Prime number theorems for Apollonian circle packings*, Senior Thesis, Princeton University (2009). - P. Sarnak,
*Letter to Lagarias on Apollonian circle packings*, http://www.math. princeton.edu/sarnak (2008).

## Bibliographic Information

**Jean Bourgain**- Affiliation: Institute for Advanced Study, School of Mathematics, Einstein Drive, Princeton, New Jersey 08540
- MR Author ID: 40280
- Email: bourgain@math.ias.edu
**Elena Fuchs**- Affiliation: Institute for Advanced Study, School of Mathematics, Einstein Drive, Princeton, New Jersey 08540
- Email: efuchs@math.ias.edu
- Received by editor(s): January 21, 2010
- Received by editor(s) in revised form: February 24, 2011, and June 6, 2011
- Published electronically: June 20, 2011
- Additional Notes: The first author is supported in part by NSF grant DMS–0808042

The second author was supported in part by NSF grant DMS–0635607 - © Copyright 2011 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**24**(2011), 945-967 - MSC (2010): Primary 11D09, 11E16, 11E20
- DOI: https://doi.org/10.1090/S0894-0347-2011-00707-8
- MathSciNet review: 2813334