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

The local-global principle for integral Soddy sphere packings

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.

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.

[2]

M. BorkovecW. 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. GrahamJ. C. LagariasC. L. MallowsA. 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. GrahamJ. C. LagariasC. L. MallowsA. 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).

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.

[2]

M. BorkovecW. 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. GrahamJ. C. LagariasC. L. MallowsA. 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. GrahamJ. C. LagariasC. L. MallowsA. 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).

Figure 3.  A reproduction from [28]
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