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

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

2010-01-21

arxiv.org/abs/1001.3894v1

Mathematics

Number Theory

20 pages

Scientific paper

A bounded 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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