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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 |
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
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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.
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R. Descartes, Œuvres, volume 4, (eds. C. Adams and P. Tannery), Paris, 1901. |
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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] | |
| [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. 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] | |
| [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). |
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