Radial density in Apollonian packings

Jayadev S. Athreya, Cristian Cobeli, Alexandru Zaharescu

Research output: Contribution to journalArticlepeer-review


Given P, an Apollonian Circle Packing, and a circle C0 =∂ B(z0, r0) in P, color the set of disks in P tangent to C0 red. What proportion of the concentric circle Cε =∂ B(z0, r0 + ε) is red, and what is the behavior of this quantity as ε→0? Using equidistribution of closed horocycles on the modular surface ℍ2/SL(2,ℤ), we show that the answer is 3/π =0.9549 . We also describe an observation due to Alex Kontorovich connecting the rate of this convergence in the Farey-Ford packing to the Riemann hypothesis. For the analogous problem for Soddy sphere packings, we find that the limiting radial density is √3/2VT =0.853 ., where VT denotes the volume of an ideal hyperbolic tetrahedron with dihedral angles π/3.

Original languageEnglish (US)
Pages (from-to)9991-10011
Number of pages21
JournalInternational Mathematics Research Notices
Issue number20
StatePublished - 2015

ASJC Scopus subject areas

  • Mathematics(all)


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