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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.

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Prime and Thickened Prime Components in Apollonian Circle Packings

  • 会议论文

  • 出版模式：

    开放获取

  • 发表日期：

    2026年

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

摘要

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.