Abstract
In this paper we study the integral properties of Apollonian 3-circle packings, which are variants of the standard Apollonian circle packings. Specifically, we study the reduction theory, formulate a local-global conjecture, and prove a density one version of this conjecture. Along the way, we prove a uniform spectral gap for the congruence towers of the symmetry group.
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-1209373
Award Identifier / Grant number: DMS-1064214
Award Identifier / Grant number: DMS-1001252
Award Identifier / Grant number: DMS-1254788
Funding statement: The author acknowledges support for this work from Prof. Kontorovich’s NSF grants DMS-1209373, DMS-1064214, DMS-1001252, and NSF CAREER grant DMS-1254788.
Acknowledgements
This paper is essentially the content of the author’s Ph.D. thesis when he was a graduate student at Stony Brook. The author is greatly indebted to his PhD advisor Prof. Alex Kontorovich for introducing this beautiful subject to the author and for numerous enlightening discussions. The author also thanks the referee for her/his numerous corrections and helpful suggestions. In writing up this paper, the author utilizes the codes provided by Prof. Kontorovich for several pictures.
References
[1] J. Bourgain and E. Fuchs, A proof of the positive density conjecture for integer Apollonian circle packings, J. Amer. Math. Soc. 24 (2011), no. 4, 945–967. 10.1090/S0894-0347-2011-00707-8Suche in Google Scholar
[2]
J. Bourgain, A. Gamburd and P. Sarnak,
Generalization of Selberg’s
[3] J. Bourgain and A. Kontorovich, On the local-global conjecture for integral Apollonian gasket, Invent. Math. 196 (2014), no. 3, 589–650, With an appendix by P. P. Varjú. 10.1007/s00222-013-0475-ySuche in Google Scholar
[4] J. Elstrodt, F. Grunewald and J. Mennicke, Groups acting on hyperbolic space, Springer, Berlin 1998. 10.1007/978-3-662-03626-6Suche in Google Scholar
[5] E. Fuchs, Arithmetic properties of Apollonian circle packings, Ph.D. thesis, Princeton University, ProQuest LLC, Ann Arbor 2010. Suche in Google Scholar
[6] A. S. Golsefidy and P. P. Varjú, Expansion in perfect groups, Geom. Funct. Anal. 22 (2012), no. 6, 1832–1891. 10.1007/s00039-012-0190-7Suche in Google Scholar
[7] 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), no. 1, 1–45. 10.1016/S0022-314X(03)00015-5Suche in Google Scholar
[8] R. L. Graham, J. C. Lagarias, C. L. Mallows, A. R. Wilks and C. H. Yan, Apollonian circle packings: Geometry and group theory. I. The Apollonian group, Discrete Comput. Geom. 34 (2005), no. 4, 547–585. 10.1007/s00454-005-1196-9Suche in Google Scholar
[9] G. Guettler and C. Mallows, A generalization of Apollonian packing of circles, J. Comb. 1 (2010), no. 1, 1–27. 10.4310/JOC.2010.v1.n1.a1Suche in Google Scholar
[10] I. Kim, Counting, mixing and equidistribution of horospheres in geometrically finite rank one locally symmetric manifolds, preprint (2011), http://arxiv.org/abs/1103.5003. 10.1515/crelle-2013-0056Suche in Google Scholar
[11]
H. D. Kloosterman,
On the representation of numbers in the form
[12] A. Kontorovich and H. Oh, Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds, J. Amer. Math. Soc. 24 (2011), no. 3, 603–648, With an appendix by H. Oh and N. Shah. 10.1090/S0894-0347-2011-00691-7Suche in Google Scholar
[13] D. G. Larman, On the Besicovitch dimension of the residual set of arbitrarily packed disks in the plane, J. London Math. Soc. 42 (1967), 292–302. 10.1112/jlms/s1-42.1.292Suche in Google Scholar
[14] P. D. Lax and R. S. Phillips, The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces, J. Funct. Anal. 46 (1982), no. 3, 280–350. 10.1016/0022-1236(82)90050-7Suche in Google Scholar
[15] S. J. Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976), no. 3–4, 241–273. 10.1007/BF02392046Suche in Google Scholar
[16] P. Sarnak, Letter to J. Lagarias about integral Apollonian packings, (2007), http://web.math.princeton.edu/sarnak/AppolonianPackings.pdf. Suche in Google Scholar
[17] F. Soddy, The bowl of integers and the hexlet, Nature 139 (1937), 77–79. 10.1038/139077a0Suche in Google Scholar
[18] D. Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math. 153 (1984), no. 3–4, 259–277. 10.1007/BF02392379Suche in Google Scholar
[19] I. Vinogradov, Effective bisector estimate with application to Apollonian circle packings, Ph.D. thesis, Princeton University, ProQuest LLC, Ann Arbor 2012. 10.1093/imrn/rnt037Suche in Google Scholar
[20] B. Weisfeiler, Strong approximation for Zariski-dense subgroups of semisimple algebraic groups, Ann. of Math. (2) 120 (1984), no. 2, 271–315. 10.2307/2006943Suche in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Limit sets of Teichmüller geodesics with minimal non-uniquely ergodic vertical foliation
- On the index of symmetric spaces
- Approximation properties for noncommutative Lp-spaces of high rank lattices and nonembeddability of expanders
- On the local-global principle for integral Apollonian 3-circle packings
- Local zero estimates and effective division in rings of algebraic power series
- Sobolev spaces and hyperbolic fillings
- Plücker varieties and higher secants of Sato’s Grassmannian
- Representation stability for homotopy groups of configuration spaces
- Counting in hyperbolic spikes: The diophantine analysis of multihomogeneous diagonal equations
Artikel in diesem Heft
- Frontmatter
- Limit sets of Teichmüller geodesics with minimal non-uniquely ergodic vertical foliation
- On the index of symmetric spaces
- Approximation properties for noncommutative Lp-spaces of high rank lattices and nonembeddability of expanders
- On the local-global principle for integral Apollonian 3-circle packings
- Local zero estimates and effective division in rings of algebraic power series
- Sobolev spaces and hyperbolic fillings
- Plücker varieties and higher secants of Sato’s Grassmannian
- Representation stability for homotopy groups of configuration spaces
- Counting in hyperbolic spikes: The diophantine analysis of multihomogeneous diagonal equations