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

Bourgain, Jean; Fuchs, Elena

doi:10.1090/S0894-0347-2011-00707-8

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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
Posted: 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 occurring as curvatures in a bounded integer ACP , and prove a conjecture of Graham, Lagarias, Mallows, Wilkes, and Yan that the ratio $\kappa(P,X)/X$ is greater than 0 for tending to infinity.

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