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

TL;DR

This paper proves that in bounded integer Apollonian circle packings, the proportion of curvatures less than X remains positive as X grows infinitely large, confirming a longstanding conjecture.

Contribution

It establishes a lower bound for the count of integer curvatures in bounded ACPs and proves the conjecture that their density ratio stays positive asymptotically.

Findings

Lower bound for the number of integer curvatures less than X

Proof that the ratio of curvatures to X remains positive as X approaches infinity

Confirmed a conjecture by Graham et al. on the density of integer curvatures

Abstract

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.