Some Experiments with Integral Apollonian Circle Packings

Abstract

Bounded Apollonian circle packings (ACP's) are constructed by repeatedly inscribing circles into the triangular interstices of a configuration of four mutually tangent circles, one of which is internally tangent to the other three. If the original four circles have integer curvature, all of the circles in the packing will have integer curvature as well. In [S1], Sarnak proves that there are infinitely many circles of prime curvature and infinitely many pairs of tangent circles of prime curvature in a primitive 1 integral ACP. In this paper, we give a heuristic backed up by numerical data for the number of circles of prime curvature less than x, and the number of "kissing primes," or pairs of circles of prime curvature less than x in a primitive integral ACP. We also provide experimental evidence towards a local to global principle for the curvatures in a primitive integral ACPs.

From Apollonius to Zaremba: Local-global phenomena in thin orbits

“…They observe empirically that congruence obstructions for any integral gasket seem to be to the modulus 24, and this is completely clarified (as we explain below) by Fuchs [Fuc11] in her thesis. Further convincing numerical evidence toward the conjecture is given in Fuch and Sanden [FS11]. Here is some recent progress.…”

Counting problems in Apollonian packings

“…Based on these experiments, they pose a "strong density conjecture" which predicts that given a primitive packing P , any sufficiently large integer satisfying some fixed congruence conditions appears as a curvature in P . This conjecture is posed in a more precise way as a local-to-global conjecture in [22]. As we discuss in Section 4, this precise local-global conjecture has stood up to experimental scrutiny and remains wide open.…”

“…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.…”

On the local-global conjecture for integral Apollonian gaskets

“…In particular, the following striking local-to-global conjecture for B is given in [GLM + 03, p. 37], [FS11]. Let A = A G denote the admissible integers, that is, those passing all local (congruence) obstructions:…”

“…[KO11,FS11,Sar11]): studying B with multiplicity (that is, studying circles), or without multiplicity (studying the integers which arise). In the present paper, we are concerned with the latter.…”