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

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 08, 2017 Revised  March 23, 2019 Published  August 2019

Fund Project: 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

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.

Keywords: Sphere packings, thin groups, hyperbolic geometry, arithmetic groups, quadratic forms, local-global principle.
Mathematics Subject Classification: Primary: 11D85; Secondary: 11F06, 20H05.
Citation: Alex Kontorovich. The local-global principle for integral Soddy sphere packings. Journal of Modern Dynamics, 2019, 15: 209-236. doi: 10.3934/jmd.2019019
References:
[1]

A. Baragar, Higher dimensional Apollonian packings, revisited, Geom. Dedicata, 195 (2018), 137-161.  doi: 10.1007/s10711-017-0280-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[7] J. W. S. Cassels, Rational Quadratic Forms, London Mathematical Society Monographs, 13, Academic Press, London-New York, 1978.   Google Scholar
[8]

R. Descartes, Œuvres, volume 4, (eds. C. Adams and P. Tannery), Paris, 1901. Google Scholar

[9]

D. Dias, The local-global principle for integral generalized Apollonian sphere packings, preprint, arXiv: 1401.4789, (2014). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

T. Gossett, The kiss precise, Nature, 139 (1937), 62.  doi: 10.1038/139062a0.  Google Scholar

[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.  Google Scholar

[15]

H. Iwaniec, Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, 17, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/gsm/017.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

R. Lachlan, On systems of circles and spheres, Philos. Roy. Soc. London Ser. A, 177 (1886), 481-625.   Google Scholar

[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.  Google Scholar

[24]

K. Nakamura, The local-global principle for integral bends in orthoplicial Apollonian sphere packings, preprint, arXiv: 1401.2980, (2014). Google Scholar

[25]

http://mathworld.wolfram.com/TangentSpheres.html. Google Scholar

[26]

P. Sarnak, Letter to J. Lagarias about integral Apollonian packings, 2007. Available from: http://web.math.princeton.edu/sarnak/AppolonianPackings.pdf. Google Scholar

[27]

F. Soddy, The kiss precise, Nature, 137 (1936), 1021.  doi: 10.1038/1371021a0.  Google Scholar

[28]

F. Soddy, The bowl of integers and the hexlet, Nature, 139 (1937), 77-79.  doi: 10.1038/139077a0.  Google Scholar

[29]

X. Zhang, On the local-global principle for integral Apollonian-3 Circle packings, preprint, arXiv: 1312.4650, (2013).  Google Scholar

show all references

References:
[1]

A. Baragar, Higher dimensional Apollonian packings, revisited, Geom. Dedicata, 195 (2018), 137-161.  doi: 10.1007/s10711-017-0280-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7] J. W. S. Cassels, Rational Quadratic Forms, London Mathematical Society Monographs, 13, Academic Press, London-New York, 1978.   Google Scholar
[8]

R. Descartes, Œuvres, volume 4, (eds. C. Adams and P. Tannery), Paris, 1901. Google Scholar

[9]

D. Dias, The local-global principle for integral generalized Apollonian sphere packings, preprint, arXiv: 1401.4789, (2014). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

T. Gossett, The kiss precise, Nature, 139 (1937), 62.  doi: 10.1038/139062a0.  Google Scholar

[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.  Google Scholar

[15]

H. Iwaniec, Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, 17, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/gsm/017.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

R. Lachlan, On systems of circles and spheres, Philos. Roy. Soc. London Ser. A, 177 (1886), 481-625.   Google Scholar

[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.  Google Scholar

[24]

K. Nakamura, The local-global principle for integral bends in orthoplicial Apollonian sphere packings, preprint, arXiv: 1401.2980, (2014). Google Scholar

[25]

http://mathworld.wolfram.com/TangentSpheres.html. Google Scholar

[26]

P. Sarnak, Letter to J. Lagarias about integral Apollonian packings, 2007. Available from: http://web.math.princeton.edu/sarnak/AppolonianPackings.pdf. Google Scholar

[27]

F. Soddy, The kiss precise, Nature, 137 (1936), 1021.  doi: 10.1038/1371021a0.  Google Scholar

[28]

F. Soddy, The bowl of integers and the hexlet, Nature, 139 (1937), 77-79.  doi: 10.1038/139077a0.  Google Scholar

[29]

X. Zhang, On the local-global principle for integral Apollonian-3 Circle packings, preprint, arXiv: 1312.4650, (2013).  Google Scholar

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