Geodesic
A proof of the positive density conjecture for integer Apollonian circle packings
Bourgain, Jean1 ; Fuchs, Elena1
1 Institute for Advanced Study, School of Mathematics, Einstein Drive, Princeton, New Jersey 08540
Journal of the American Mathematical Society, Tome 24 (2011) no. 4, pp. 945-967

Voir la notice de l'article provenant de la source American Mathematical Society

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.
DOI : 10.1090/S0894-0347-2011-00707-8

Bourgain, Jean 1 ; Fuchs, Elena 1

1 Institute for Advanced Study, School of Mathematics, Einstein Drive, Princeton, New Jersey 08540 
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Bourgain, Jean; Fuchs, Elena. A proof of the positive density conjecture for integer Apollonian circle packings. Journal of the American Mathematical Society, Tome 24 (2011) no. 4, pp. 945-967. doi: 10.1090/S0894-0347-2011-00707-8

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