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Counting problems in Apollonian packings


Author: Elena Fuchs
Journal: Bull. Amer. Math. Soc. 50 (2013), 229-266
MSC (2010): Primary 11-02
DOI: https://doi.org/10.1090/S0273-0979-2013-01401-0
Published electronically: February 14, 2013
Previous version: Original version posted January 16, 2013
Corrected version: Current version includes corrected Figure 2 and updated subject classification.
MathSciNet review: 3020827
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Abstract: An Apollonian circle packing is a classical construction which is made by repeatedly inscribing circles into the triangular interstices in a Descartes configuration of four mutually tangent circles. Remarkably, if the original four circles have integer curvature, all of the circles in the packing will have integer curvature, making the packings of interest from a number theoretic point of view. Many of the natural arithmetic problems have required new and sophisticated tools to solve them. The reason for this difficulty is that the study of Apollonian packings reduces to the study of a subgroup of $ \textrm {GL}_4(\mathbb{Z})$ that is thin in a sense that we describe in this article, and arithmetic problems involving thin groups have only recently become approachable in broad generality. In this article, we report on what is currently known about Apollonian packings in which all circles have integer curvature and how these results are obtained. This survey is also meant to illustrate how to treat arithmetic problems related to other thin groups.


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Additional Information

Elena Fuchs
Affiliation: Department of Mathematics, University of California, Berkeley, California
Email: efuchs@math.berkeley.edu

DOI: https://doi.org/10.1090/S0273-0979-2013-01401-0
Received by editor(s): August 1, 2012
Received by editor(s) in revised form: September 2, 2012
Published electronically: February 14, 2013
Additional Notes: The author is supported by the Simons Foundation through the Postdoctoral Fellows program
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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