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

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

Quantitative ergodic theorems and their number-theoretic applications

“…Other results in this rapidly developing area include the extensive development regarding Apollonian packings [S07,S09,KO11,BF11] and orbits of thin subgroups more generally [K09,BK10,KO12]. We refer to [K, O11, O14] for recent comprehensive surveys on this subject.…”

Counting problems in Apollonian packings

“…For example, if P is taken to be the packing generated by (−1, 2, 2, 3), it is shown that P 24 = {2, 3, 6, 11, 14, 15, 18, 23} and that all integers 10 6 < x < 5 · 10 8 such that x ∈ P 24 modulo 24 appear as curvatures in P . An immediate consequence of Conjecture 4.1 is the positive density conjecture of Graham et al in [27] that the curvatures in a given packing have positive density in N which was first proven in [8]. In this section we outline the proof of this positive density conjecture and survey what is currently known about this density and about the local-to-global conjecture above.…”

“…Indeed, this remarkable integrality feature gives rise to several natural questions about integer ACPs; Graham et al make some progress towards answering them in [27] and pose striking conjectures, many of which are now theorems or at least better understood (see [7], [8], [9], [13], [19], [20], [21], [22], [32], [46], etc.) In this article we will survey how all these questions are handled and give an overview of what is currently known.…”

“…The latter involves counting curvatures appearing in a packing with multiplicity, rather than counting every integer that comes up exactly once, as is done in [8] and summarized in Section 4 of this article.…”

“…In [8], this Fuchsian subgroup method was enhanced in a number of ways to settle Question 3 and to prove Theorem 4.2 below, that κ(P, X) X, where the implied constant depends on the packing P . Recently, a further refinement of this analysis coupled with new techniques, introducing the circle method for thin orbits ( [12], [13]) as well as the congruence analysis in [21], has given asymptotics for κ(P, X) as X → ∞ (see Theorem 4.3 and the discussion in Section 4).…”