The local-global principle for integral Soddy sphere packings

Alex Kontorovich

doi:10.3934/jmd.2019019

Article Contents

2019, Volume 15: 209-236. Doi: 10.3934/jmd.2019019

This volume Previous Article Mather theory and symplectic rigidity Next Article Counting saddle connections in a homology class modulo

$ \boldsymbol q $

(with an appendix by Rodolfo Gutiérrez-Romo)

The local-global principle for integral Soddy sphere packings

Alex Kontorovich

Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd., Piscataway, NJ 08854, USA

Received: November 07, 2017

Revised: March 22, 2019

Published: August 2019

The author is partially supported by an NSF CAREER grant DMS-1254788 and DMS-1455705, an NSF FRG grant DMS-1463940, an Alfred P. Sloan Research Fellowship, and a BSF grant

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Abstract

Fix an integral Soddy sphere packing $ \mathscr{P} $. Let $ \mathscr{B} $ be the set of all bends in $ \mathscr{P} $. A number $ n $ is called represented if $ n\in \mathscr{B} $, that is, if there is a sphere in $ \mathscr{P} $ with bend equal to $ n $. A number $ n $ is called admissible if it is everywhere locally represented, meaning that $ n\in \mathscr{B}( \operatorname{mod} q) $ for all $ q $. It is shown that every sufficiently large admissible number is represented.

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Mathematics Subject Classification: Primary: 11D85; Secondary: 11F06, 20H05.

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Figure 3. A reproduction from [28]

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References

[1]	A. Baragar, Higher dimensional Apollonian packings, revisited, Geom. Dedicata, 195 (2018), 137-161. doi: 10.1007/s10711-017-0280-7.
[2]	M. Borkovec, W. de Paris and R. Peikert, The fractal dimension of the Apollonian sphere packing, Fractals, 2 (1994), 521-526. doi: 10.1142/S0218348X94000739.
[3]	J. Bourgain and E. Fuchs, A proof of the positive density conjecture for integer Apollonian circle packings, J. Amer. Math. Soc., 24 (2011), 945-967. doi: 10.1090/S0894-0347-2011-00707-8.
[4]	J. Bourgain and A. Kontorovich, On the local-global conjecture for integral Apollonian gaskets, Invent. Math., 196 (2014), 589-650. doi: 10.1007/s00222-013-0475-y.
[5]	D. W. Boyd, An algorithm for generating the sphere coordinates in a three-dimensional osculatory packing, Math. Comp., 27 (1973), 369-377. doi: 10.1090/S0025-5718-1973-0338937-6.
[6]	D. W. Boyd, The osculatory packing of a three dimensional sphere, Can. J. Math., 25 (1973), 303-322. doi: 10.4153/CJM-1973-030-5.
[7]	J. W. S. Cassels, Rational Quadratic Forms, London Mathematical Society Monographs, 13, Academic Press, London-New York, 1978.
[8]	R. Descartes, Œuvres, volume 4, (eds. C. Adams and P. Tannery), Paris, 1901.
[9]	D. Dias, The local-global principle for integral generalized Apollonian sphere packings, preprint, arXiv: 1401.4789, (2014).
[10]	E. Fuchs and K. Sanden, Some experiments with integral Apollonian circle packings, Exp. Math., 20 (2011), 380-399. doi: 10.1080/10586458.2011.565255.
[11]	R. L. Graham, J. C. Lagarias, C. L. Mallows, A. R. Wilks and C. H. Yan, Apollonian circle packings: Number theory, J. Number Theory, 100 (2003), 1-45. doi: 10.1016/S0022-314X(03)00015-5.
[12]	R. L. Graham, J. C. Lagarias, C. L. Mallows, A. R. Wilks and C. H. Yan, Apollonian circle packings: Geometry and group theory. Ⅲ. Higher dimensions, Discrete Comput. Geom., 35 (2006), 37-72. doi: 10.1007/s00454-005-1197-8.
[13]	T. Gossett, The kiss precise, Nature, 139 (1937), 62. doi: 10.1038/139062a0.
[14]	F. Grunewald and J. Schwermer, Subgroups of Bianchi groups and arithmetic quotients of hyperbolic 3-space, Trans. Amer. Math. Soc., 335 (1993), 47-78. doi: 10.2307/2154257.
[15]	H. Iwaniec, Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, 17, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/gsm/017.
[16]	I. Kim, Counting, mixing and equidistribution of horospheres in geometrically finite rank one locally symmetric manifolds, J. Reine Angew. Math., 704 (2015), 85-133. doi: 10.1515/crelle-2013-0056.
[17]	H. D. Kloosterman, On the representation of numbers in the form ax²+by²+ cz²+dt², Acta Math., 49 (1927), 407-464. doi: 10.1007/BF02564120.
[18]	A. Kontorovich and K. Nakamura, Geometry and arithmetic of crystallographic sphere packings, Proc. Natl. Acad. Sci. USA, 116 (2019), 436-441. doi: 10.1073/pnas.1721104116.
[19]	A. Kontorovich and H. Oh, Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds, J. Amer. Math. Soc., 24 (2011), 603-648. doi: 10.1090/S0894-0347-2011-00691-7.
[20]	A. Kontorovich, From Apollonius to Zaremba: Local-global phenomena in thin orbits, Bull. Amer. Math. Soc. (N.S.), 50 (2013), 187-228. doi: 10.1090/S0273-0979-2013-01402-2.
[21]	A. Kontorovich, Applications of thin orbits, in Dynamics and Analytic Number Theory, London Math. Soc. Lecture Note Ser., 437, Cambridge Univ. Press, Cambridge, 2016, 289–317.
[22]	R. Lachlan, On systems of circles and spheres, Philos. Roy. Soc. London Ser. A, 177 (1886), 481-625.
[23]	J. Milnor, Hyperbolic geometry: The first 150 years, Bull. Amer. Math. Soc. (N.S.), 6 (1982), 9-24. doi: 10.1090/S0273-0979-1982-14958-8.
[24]	K. Nakamura, The local-global principle for integral bends in orthoplicial Apollonian sphere packings, preprint, arXiv: 1401.2980, (2014).
[25]	http://mathworld.wolfram.com/TangentSpheres.html.
[26]	P. Sarnak, Letter to J. Lagarias about integral Apollonian packings, 2007. Available from: http://web.math.princeton.edu/sarnak/AppolonianPackings.pdf.
[27]	F. Soddy, The kiss precise, Nature, 137 (1936), 1021. doi: 10.1038/1371021a0.
[28]	F. Soddy, The bowl of integers and the hexlet, Nature, 139 (1937), 77-79. doi: 10.1038/139077a0.
[29]	X. Zhang, On the local-global principle for integral Apollonian-3 Circle packings, preprint, arXiv: 1312.4650, (2013).