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

By Jean Bourgain, E. Fuchs. 2010

71 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/9711a075166dc5e6ddeb60d8bbadc97661437f20

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

On the local-global conjecture for integral Apollonian gaskets (2012, methodology): “], who showed that #(B \[1;N]) ˛N1=2: (1.5) Sarnak [Sar07] improved this to #(B \[1;N]) ˛ N (logN)1=2 ; (1.6) and then Fuchs [Fuc10] showed #(B \[1;N]) ˛ N (logN)0:150::: : Finally Bourgain and Fuchs [BF10] settled the so-called \Positive Density Conjecture," that #(B \[1;N]) ˛N: 4 JEAN BOURGAI”
From Apollonius To Zaremba: Local-Global Phenomena in Thin Orbits (2012, background): “e history on this problem: Graham et al [GLM+03] already made the rst progress, proving that #(B \[1;N]) ˛N1=2: (3.25) Then Sarnak [Sar07] showed #(B \[1;N]) ˛ N p logN ; (3.26) before Bourgain-Fuchs [BF11] settled the so-called Positive Density Conjecture, that #(B \[1;N]) ˛N: (3.27) A key observat”
Notes on Thin Matrix Groups (2012, methodology): “In [Bourgain and Kontorovich 2013b] the theory of thin subgroups of SL2(Z) is extended to thin semi(sub)groups (one has to abandon direct spectral methods and replace them by dynamical ones [Lalley 1989; Bourgain et al. 2011]).”
Expander graphs in pure and applied mathematics (2011, background): “re innitely many twin prime? i.e., are there innitely many kissing pairs of circles with prime curvatures. A rich theory has started to emerge in recent years (cf. [GLMWY], [S5], [S8], [KO1], [Fu], [BF], [FS] and the reference therein). This is a fascinating crossroad of number theory, geometry, gro”