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

Bourgain, Jean; Fuchs, Elena

doi:10.1090/S0894-0347-2011-00707-8

A proof of the positive density conjecture for integer Apollonian circle packings
HTML articles powered by AMS MathViewer

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

PDF | Request permission

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).