An Apollonian circle packing is a fractal arrangement formed by repeatedly inscribing circles into the interstices in a Descartes configuration of four mutually tangent circles. The curvatures of the circles in such a packing are often integers, and so it is natural to ask questions about their arithmetic properties. For example, it is known by work of Bourgain-Fuchs that a positive fraction of integers appear as curvatures in any integral Apollonian circle packing. In this talk, we investigate the arithmetic properties of the collection of integers appearing in thickened prime components of Apollonian circle packings.
Prime components in Apollonian packings
Catherine Hsu, Swarthmore CollegeAuthors: Holley Friedlander, Elena Fuchs, Piper H, Catherine Hsu, Katherine Sanden, Damaris Schindler, and Katherine Strange
2022 AWM Research Symposium
New Directions in Number Theory