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Prime and thickened prime components in Apollonian circle packings

Holley Friedlander, Elena Fuchs, Piper Harris, Catherine Hsu, James Rickards, Katherine Sanden, Damaris Schindler, Katherine E. Stange

TL;DR

This work introduces and analyzes prime components and thickened prime components within primitive integral Apollonian circle packings, exploring local obstructions, residue classes, and growth behavior of curvatures. It leverages Sarnak's link between curvatures and primes represented by shifted binary quadratic forms, along with pinch-family and Cayley-graph techniques, to obtain local results and two-layer lower bounds for integers represented by thickened prime components. The authors prove infinite growth for prime components, provide conjectures for the number of such components (PCR), and establish a nontrivial lower bound $N^{\text{th}}(X) \gg X/(\log\log X)^{1/2}$ for integers up to $X$ appearing in thickened prime components, conditional on standard conjectures. Extensive computational data up to curvature ~ $10^{13}$ supports the proposed growth rates and residue-class behavior, and the work formulates open questions about extending these results to broader settings and higher thickenings.

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.

Prime and thickened prime components in Apollonian circle packings

TL;DR

This work introduces and analyzes prime components and thickened prime components within primitive integral Apollonian circle packings, exploring local obstructions, residue classes, and growth behavior of curvatures. It leverages Sarnak's link between curvatures and primes represented by shifted binary quadratic forms, along with pinch-family and Cayley-graph techniques, to obtain local results and two-layer lower bounds for integers represented by thickened prime components. The authors prove infinite growth for prime components, provide conjectures for the number of such components (PCR), and establish a nontrivial lower bound for integers up to appearing in thickened prime components, conditional on standard conjectures. Extensive computational data up to curvature ~ supports the proposed growth rates and residue-class behavior, and the work formulates open questions about extending these results to broader settings and higher thickenings.

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 ; software is available.
Paper Structure (40 sections, 39 theorems, 99 equations, 17 figures)

This paper contains 40 sections, 39 theorems, 99 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Local-to-global for Apollonian packings
  3. Local results.
  4. Size of prime components
  5. Counting integers in prime components
  6. Counting prime components
  7. Experimental data.
  8. A note on the images and software.
  9. Paper structure.
  10. Acknowledgements.
  11. Background
  12. Descartes quadruples and the Apollonian group
  13. Local properties
  14. Quadratic families
  15. Pinch families
  16. ...and 25 more sections

Key Result

Theorem 1.0.3

Let ${\mathcal{P}^{\operatorname{th}}}$ be a thickened prime component in a primitive integral Apollonian packing. Then the number $N_{\mathcal{P}^{\operatorname{th}}}(X)$ of distinct integers less than $X$ occuring in ${\mathcal{P}^{\operatorname{th}}}$ satisfies

Figures (17)

  • Figure 1: The construction of an Apollonian packing
  • Figure 2: Circles of curvature $\leq 30000$ in the Apollonian circle packing corresponding to $(-47, 97, 100, 108)$.
  • Figure 3: The packing informally known as the People's packing, up to curvature 10,000. In pink, a prime component. In blue, the additional circles in the thickened prime component. All other circles are purple.
  • Figure 4: A quadratic family (all circles tangent to the circle of curvature 11) in the People's packing. The quadratic family is shown in grey. For circles not in the quadratic family, colours indicate residue class modulo $7$. Font colour distinguishes primes and composites.
  • Figure 5: A pinch family in the People's packing. The pinch family is shown in grey. For circles not in the pinch family, colours indicate residue class modulo $7$. Font colour distinguishes primes and composites.
  • ...and 12 more figures

Theorems & Definitions (79)

  • Conjecture 1.0.1: Apollonian curvature conjecture HKRS, modifying GLMWY,FuchsSanden
  • Conjecture 1.0.2
  • Theorem 1.0.3
  • Theorem 1.1.1: Sarnak
  • Theorem 1.1.2: BourgainFuchs
  • Theorem 1.1.3: BourgainKontorovich
  • Theorem : Theorem \ref{['thm:primeclasses']}
  • Theorem : Theorem \ref{['thm:2-step walk-prime']}
  • Conjecture 1.3.1
  • Conjecture : Conjecture \ref{['PCRconjecture']}
  • ...and 69 more