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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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A proof of the positive density conjecture for integer Apollonian circle packings
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by Jean Bourgain and Elena Fuchs PDF
J. Amer. Math. Soc. 24 (2011), 945-967 Request permission

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.
References
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Additional Information
  • Jean Bourgain
  • Affiliation: Institute for Advanced Study, School of Mathematics, Einstein Drive, Princeton, New Jersey 08540
  • MR Author ID: 40280
  • 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
  • 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
  • © Copyright 2011 American Mathematical Society
  • 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
  • MathSciNet review: 2813334