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A proof of the positive density conjecture for integer Apollonian circle packings


Authors: Jean Bourgain and Elena Fuchs
Journal: J. Amer. Math. Soc. 24 (2011), 945-967
MSC (2010): Primary 11D09, 11E16, 11E20
DOI: https://doi.org/10.1090/S0894-0347-2011-00707-8
Published electronically: June 20, 2011
MathSciNet review: 2813334
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Abstract: An Apollonian circle packing (ACP) is an ancient Greek 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 as well. In this paper, we compute a lower bound for the number $ \kappa(P,X)$ of integers less than $ X$ occurring as curvatures in a bounded integer ACP $ P$, and prove a conjecture of Graham, Lagarias, Mallows, Wilkes, and Yan that the ratio $ \kappa(P,X)/X$ is greater than 0 for $ X$ tending to infinity.


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Additional Information

Jean Bourgain
Affiliation: Institute for Advanced Study, School of Mathematics, Einstein Drive, Princeton, New Jersey 08540
Email: bourgain@math.ias.edu

Elena Fuchs
Affiliation: Institute for Advanced Study, School of Mathematics, Einstein Drive, Princeton, New Jersey 08540
Email: efuchs@math.ias.edu

DOI: https://doi.org/10.1090/S0894-0347-2011-00707-8
Keywords: Apollonian packings, number theory, quadratic forms, sieve methods, circle method
Received by editor(s): January 21, 2010
Received by editor(s) in revised form: February 24, 2011, and June 6, 2011
Published electronically: June 20, 2011
Additional Notes: The first author is supported in part by NSF grant DMS–0808042
The second author was supported in part by NSF grant DMS–0635607
Article copyright: © Copyright 2011 American Mathematical Society

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