Prime and thickened prime components in Apollonian circle packings

TL;DR

This paper introduces the concept of prime and thickened prime components in Apollonian circle packings, analyzing their curvature properties, residue classes, and distribution through extensive computational experiments.

Contribution

It defines prime components in Apollonian packings and investigates their properties, including residue classes and distribution, supported by large-scale computational analysis.

Findings

Prime components have specific residue class distributions.

Number of prime components varies across packings.

Curvature sets exhibit particular statistical properties.

Abstract

Inspired by a question of Sarnak, we introduce the notion of a prime component in an Apollonian circle packing: a maximal tangency-connected subset having all prime curvatures. We also consider thickened prime components, which are augmented by all circles immediately tangent to the prime component. In both cases, we ask about the curvatures which appear. We consider the residue classes attained by the set of curvatures, the number of circles in such components, the number of distinct integers occurring as curvatures, and the number of prime components in a packing. As part of our investigation, we computed and analysed example components up to around curvature $10^{13}$; software is available.