Abstract
A family of fractal arrangements of circles is introduced for each imaginary quadratic field K. Collectively, these arrangements contain (up to an affine transformation) every set of circles in the extended complex plane with integral curvatures and Zariski dense symmetry group. When that set is a circle packing, we show how the ambient structure of our arrangement gives a geometric criterion for satisfying the asymptotic local–global principle. Connections to the class group of K are also explored. Among them is a geometric property that guarantees certain ideal classes are group generators.
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References
Baragar, A., Lautzenheiser, D.: A game of packings. arXiv:2006.00619 (2020)
Bogachev, N., Kolpakov, A., Kontorovich, A.: Kleinian sphere packings, reflection groups, and arithmeticity. arXiv:2203.01973 (2021)
Bourgain, J., Kontorovich, A.: On the local-global conjecture for integral Apollonian gaskets. Invent. Math. 196(3), 589–650 (2014)
Boyd, D.: A new class of infinite sphere packings. Pac. J. Math. 50(2), 383–398 (1974)
Carter, R.W., Segal, G.B., MacDonald, I.G.: Lectures on Lie Groups and Lie Algebras, vol. 32. Cambridge University Press, Cambridge (1995)
Chait-Roth, D., Cui, A., Stier, Z.: A taxonomy of crystallographic sphere packings. J. Number Theory 207, 196–246 (2020)
Elstrodt, J., Grunewald, F., Mennicke, J.: Groups Acting on Hyperbolic Space: Harmonic Analysis and Number Theory. Springer, Berlin (2013)
Fuchs, E.: Strong approximation in the Apollonian group. J. Number Theory 131(12), 2282–2302 (2011)
Fuchs, E., Stange, K.E., Zhang, X.: Local-global principles in circle packings. Compos. Math. 155(6), 1118–1170 (2019)
Graham, R.L., Lagarias, J.C., Mallows, C.L., Wilks, A.R., Yan, C.H.: Apollonian circle packings: number theory. J. Number Theory 100(1), 1–45 (2003)
Graham, R.L., Lagarias, J.C., Mallows, C.L., Wilks, A.R., Yan, C.H.: Apollonian circle packings: geometry and group theory I: the Apollonian group. Discrete Comput. Geom. 34(4), 547–585 (2005)
Graham, R.L., Lagarias, J.C., Mallows, C.L., Wilks, A.R., Yan, C.H.: Apollonian circle packings: geometry and group theory II. Super-Apollonian group and integral packings. Discrete & Computational Geometry 35(1), 1–36 (2006)
Graham, R.L., Lagarias, J.C., Mallows, C.L., Wilks, A.R., Yan, C.H.: Apollonian circle packings: geometry and group theory III. Higher dimensions. Discrete Comput. Geom. 35(1), 37–72 (2006)
Guettler, G., Mallows, C.: A generalization of Apollonian packing of circles. J. Comb. 1(1), 1–27 (2010)
Haag, S., Kertzer, C., Rickards, J., Stange, K.E.: The Local-Global Conjecture for Apollonian circle packings is false. arXiv:2307.02749 (2023)
Kapovich, M., Kontorovich, A.: On Superintegral Kleinian Sphere Packings, Bugs, and Arithmetic Groups. arXiv:2104.13838 (2021)
Kontorovich, A., Nakamura, K.: Geometry and arithmetic of crystallographic sphere packings. Proc. Natl. Acad. Sci. 116(2), 436–441 (2019)
Mackenzie, D.: A tisket, a tasket, an Apollonian gasket. Am. Sci. 98(1), 10–14 (2010)
Maclachlan, C., Reid, A.W.: The Arithmetic of Hyperbolic 3-Manifolds, vol. 219. Springer, New York (2003)
Manna, S.S., Herrmann, H.J.: Precise determination of the fractal dimensions of Apollonian packing and space-filling bearings. J. Phys. A: Math. Gen. 24(9), 481–490 (1991)
Martin, D.E.: The geometry of imaginary quadratic fields. PhD thesis, University of Colorado at Boulder (2020)
Martin, D.E.: Continued fractions over non-Euclidean imaginary quadratic rings. J. Number Theory 243, 688–714 (2022)
Niven, I.: Irrational Numbers. Mathematical Association of America, Washington, DC (1956)
Rapinchuk, A.S.: Strong approximation for algebraic groups. Math. Sci. Res. Inst. Publ. 61, 269–298 (2014)
Sarnak, P.: Integral Apollonian packings. Am. Math. Mon. 118(4), 291–306 (2011)
Soddy, F.: The bowl of integers and the hexlet. Nature 139(3506), 77–79 (1937)
Stange, K.E.: Visualizing the arithmetic of imaginary quadratic fields. Int. Math. Res. Not. 2018(12), 3908–3938 (2017)
Stange, K.E.: The Apollonian structure of Bianchi groups. Trans. Am. Math. Soc. 370, 6169–6219 (2018)
Swan, R.G.: Generators and relations for certain special linear groups. Adv. Math. 6(1), 1–77 (1971)
Acknowledgements
The author is grateful to Elena Fuchs and Katherine Stange for their guidance and support, to Katherine Stange and Xin Zhang for verifying the switch to \(\text {PGL}\) in their theorem, to Alex Kontorovich for helping with connections to Kleinian packings/bugs, and to Robert Hines for many helpful conversations.
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Martin, D.E. A Geometric Study of Circle Packings and Ideal Class Groups. Discrete Comput Geom (2024). https://doi.org/10.1007/s00454-024-00638-w
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DOI: https://doi.org/10.1007/s00454-024-00638-w