Counting problems in Apollonian packings
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Abstract:
An Apollonian circle packing is a classical 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, making the packings of interest from a number theoretic point of view. Many of the natural arithmetic problems have required new and sophisticated tools to solve them. The reason for this difficulty is that the study of Apollonian packings reduces to the study of a subgroup of $\textrm {GL}_4(\mathbb Z)$ that is thin in a sense that we describe in this article, and arithmetic problems involving thin groups have only recently become approachable in broad generality. In this article, we report on what is currently known about Apollonian packings in which all circles have integer curvature and how these results are obtained. This survey is also meant to illustrate how to treat arithmetic problems related to other thin groups.References
- D. Aharonov and K. Stephenson, Geometric sequences of discs in the Apollonian packing, Algebra i Analiz 9 (1997), no. 3, 104â140; English transl., St. Petersburg Math. J. 9 (1998), no. 3, 509â542. MR 1466797
- N. Alon, Eigenvalues and expanders, Combinatorica 6 (1986), no. 2, 83â96. Theory of computing (Singer Island, Fla., 1984). MR 875835, DOI 10.1007/BF02579166
- N. Alon and V. D. Milman, $\lambda _1,$ isoperimetric inequalities for graphs, and superconcentrators, J. Combin. Theory Ser. B 38 (1985), no. 1, 73â88. MR 782626, DOI 10.1016/0095-8956(85)90092-9
- Richard Aoun, Random subgroups of linear groups are free, Duke Math. J. 160 (2011), no. 1, 117â173. MR 2838353, DOI 10.1215/00127094-1443493
- P. Bernays, Ăber die Darstellung von positiven, ganzen Zahlen durch die primitiven, binĂ€ren quadratischen Formen einer nicht quadratischen Diskriminante, Ph.D. dissertation, Georg-August-UniversitĂ€t, Göttingen, Germany (1912).
- Valentin Blomer and Andrew Granville, Estimates for representation numbers of quadratic forms, Duke Math. J. 135 (2006), no. 2, 261â302. MR 2267284, DOI 10.1215/S0012-7094-06-13522-6
- J. Bourgain, Integral Apollonian circle packings and prime curvatures, J. Anal. Math. 118 (2012), no. 1, 221â249. MR 2993027, DOI 10.1007/s11854-012-0034-2
- Jean Bourgain and Elena Fuchs, A proof of the positive density conjecture for integer Apollonian circle packings, J. Amer. Math. Soc. 24 (2011), no. 4, 945â967. MR 2813334, DOI 10.1090/S0894-0347-2011-00707-8
- J. Bourgain and E. Fuchs, On representation of integers by binary quadratic forms, Int. Math. Res. Not., doi: 10.1093/imrn/rnr253 (2012).
- Jean Bourgain, Alex Gamburd, and Peter Sarnak, Affine linear sieve, expanders, and sum-product, Invent. Math. 179 (2010), no. 3, 559â644. MR 2587341, DOI 10.1007/s00222-009-0225-3
- Jean Bourgain, Alex Gamburd, and Peter Sarnak, Generalization of Selbergâs $\frac {3}{16}$ theorem and affine sieve, Acta Math. 207 (2011), no. 2, 255â290. MR 2892611, DOI 10.1007/s11511-012-0070-x
- Jean Bourgain and Alex Kontorovich, On representations of integers in thin subgroups of $\textrm {SL}_2(\Bbb Z)$, Geom. Funct. Anal. 20 (2010), no. 5, 1144â1174. MR 2746949, DOI 10.1007/s00039-010-0093-4
- J. Bourgain and A. Kontorovich, On the strong density conjecture for integral Apollonian circle packings, arXiv:1205.4416v1 (2012).
- David W. Boyd, The sequence of radii of the Apollonian packing, Math. Comp. 39 (1982), no. 159, 249â254. MR 658230, DOI 10.1090/S0025-5718-1982-0658230-7
- Jing Run Chen, On the representation of a larger even integer as the sum of a prime and the product of at most two primes, Sci. Sinica 16 (1973), 157â176. MR 434997
- Giuliana Davidoff, Peter Sarnak, and Alain Valette, Elementary number theory, group theory, and Ramanujan graphs, London Mathematical Society Student Texts, vol. 55, Cambridge University Press, Cambridge, 2003. MR 1989434, DOI 10.1017/CBO9780511615825
- 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
- Jordan S. Ellenberg, Chris Hall, and Emmanuel Kowalski, Expander graphs, gonality, and variation of Galois representations, Duke Math. J. 161 (2012), no. 7, 1233â1275. MR 2922374, DOI 10.1215/00127094-1593272
- E. Fuchs, A note on the density of Apollonian curvatures in $\mathbb Z$, http://math.berkeley.edu/ Ëefuchs/posdensapollo.pdf (2009).
- Elena Fuchs, Arithmetic properties of Apollonian circle packings, ProQuest LLC, Ann Arbor, MI, 2010. Thesis (Ph.D.)âPrinceton University. MR 2941628
- Elena Fuchs, Strong approximation in the Apollonian group, J. Number Theory 131 (2011), no. 12, 2282â2302. MR 2832824, DOI 10.1016/j.jnt.2011.05.010
- Elena Fuchs and Katherine Sanden, Some experiments with integral Apollonian circle packings, Exp. Math. 20 (2011), no. 4, 380â399. MR 2859897, DOI 10.1080/10586458.2011.565255
- Alex Gamburd, On the spectral gap for infinite index âcongruenceâ subgroups of $\textrm {SL}_2(\mathbf Z)$, Israel J. Math. 127 (2002), 157â200. MR 1900698, DOI 10.1007/BF02784530
- 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, 547â585. MR 2173929, DOI 10.1007/s00454-005-1196-9
- Ronald L. Graham, Jeffrey C. Lagarias, Colin L. Mallows, Allan R. Wilks, and Catherine H. Yan, Apollonian circle packings: geometry and group theory. II. Super-Apollonian group and integral packings, Discrete Comput. Geom. 35 (2006), no. 1, 1â36. MR 2183489, DOI 10.1007/s00454-005-1195-x
- Ronald L. Graham, Jeffrey C. Lagarias, Colin L. Mallows, Allan R. Wilks, and Catherine H. Yan, Apollonian circle packings: geometry and group theory. III. Higher dimensions, Discrete Comput. Geom. 35 (2006), no. 1, 37â72. MR 2183490, DOI 10.1007/s00454-005-1197-8
- 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, 1â45. MR 1971245, DOI 10.1016/S0022-314X(03)00015-5
- Chris Hall, Big symplectic or orthogonal monodromy modulo $l$, Duke Math. J. 141 (2008), no. 1, 179â203. MR 2372151, DOI 10.1215/S0012-7094-08-14115-8
- K. E. Hirst, The Apollonian packing of circles, J. London Math. Soc. 42 (1967), 281â291. MR 209981, DOI 10.1112/jlms/s1-42.1.281
- Shlomo Hoory, Nathan Linial, and Avi Wigderson, Expander graphs and their applications, Bull. Amer. Math. Soc. (N.S.) 43 (2006), no. 4, 439â561. MR 2247919, DOI 10.1090/S0273-0979-06-01126-8
- B. Huppert, Endliche Gruppen. I, Die Grundlehren der mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703, DOI 10.1007/978-3-642-64981-3
- Alex Kontorovich and Hee Oh, Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds, J. Amer. Math. Soc. 24 (2011), no. 3, 603â648. With an appendix by Oh and Nimish Shah. MR 2784325, DOI 10.1090/S0894-0347-2011-00691-7
- E. Kowalski, Sieve in expansion, SĂ©minaire Bourbaki No. 1028 (2011).
- M. Lee and H. Oh, Effective circle count for Apollonian packings and closed horospheres, GAFA, to appear (2012).
- Alexander Lubotzky, Discrete groups, expanding graphs and invariant measures, Progress in Mathematics, vol. 125, BirkhÀuser Verlag, Basel, 1994. With an appendix by Jonathan D. Rogawski. MR 1308046, DOI 10.1007/978-3-0346-0332-4
- G. A. Margulis, Explicit constructions of expanders, Problemy PeredaÄi Informacii 9 (1973), no. 4, 71â80 (Russian). MR 0484767
- G. A. Margulis, Explicit constructions of graphs without short cycles and low density codes, Combinatorica 2 (1982), no. 1, 71â78. MR 671147, DOI 10.1007/BF02579283
- Curtis T. McMullen, Hausdorff dimension and conformal dynamics. III. Computation of dimension, Amer. J. Math. 120 (1998), no. 4, 691â721. MR 1637951, DOI 10.1353/ajm.1998.0031
- Z. A. Melzak, Infinite packings of disks, Canadian J. Math. 18 (1966), 838â852. MR 203594, DOI 10.4153/CJM-1966-084-8
- N. Niedermowwe, The circle method with weights for the representation of integers by quadratic forms, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 377 (2010), no. Issledovaniya po Teorii Chisel. 10, 91â110, 243 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 171 (2010), no. 6, 753â764. MR 2753652, DOI 10.1007/s10958-010-0180-y
- V. V. Nikulin, Discrete reflection groups in Lobachevsky spaces and algebraic surfaces, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 654â671. MR 934268
- Hee Oh and Nimish Shah, The asymptotic distribution of circles in the orbits of Kleinian groups, Invent. Math. 187 (2012), no. 1, 1â35. MR 2874933, DOI 10.1007/s00222-011-0326-7
- A. Salehi-Golsefidy, P. Sarnak, Affine sieve, arXiv:1109.6432v1 (2011).
- A. Salehi Golsefidy and PĂ©ter P. VarjĂș, Expansion in perfect groups, Geom. Funct. Anal. 22 (2012), no. 6, 1832â1891. MR 3000503, DOI 10.1007/s00039-012-0190-7
- Peter Sarnak, Integral Apollonian packings, Amer. Math. Monthly 118 (2011), no. 4, 291â306. MR 2800340, DOI 10.4169/amer.math.monthly.118.04.291
- P. Sarnak, Letter to Lagarias, http://www.math.princeton.edu/sarnak (2007).
- P. Sarnak, Notes on thin groups, MSRI Hot Topics Workshop, http://www.msri.org/ attachments/workshops/652_Sarnak-notes.pdf (2012).
- Jean-Pierre Serre, Abelian $l$-adic representations and elliptic curves, 2nd ed., Advanced Book Classics, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989. With the collaboration of Willem Kuyk and John Labute. MR 1043865
- F. Soddy, The Kiss Precise, Nature 137 No. 1021 (1936).
- I. Vinogradov, Effective bisector estimate with application to Apollonian circle packings, Princeton University Thesis (2012).
- Boris Weisfeiler, Strong approximation for Zariski-dense subgroups of semisimple algebraic groups, Ann. of Math. (2) 120 (1984), no. 2, 271â315. MR 763908, DOI 10.2307/2006943
- J. B. Wilker, Inversive geometry, The geometric vein, Springer, New York-Berlin, 1981, pp. 379â442. MR 661793
Additional Information
- Elena Fuchs
- Affiliation: Department of Mathematics, University of California, Berkeley, California
- Email: efuchs@math.berkeley.edu
- Received by editor(s): August 1, 2012
- Received by editor(s) in revised form: September 2, 2012
- Published electronically: February 14, 2013
- Additional Notes: The author is supported by the Simons Foundation through the Postdoctoral Fellows program
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Bull. Amer. Math. Soc. 50 (2013), 229-266
- MSC (2010): Primary 11-02
- DOI: https://doi.org/10.1090/S0273-0979-2013-01401-0
- MathSciNet review: 3020827