On the Local-Global Conjecture for integral Apollonian gaskets

Abstract

We prove that a set of density one satisfies the local-global conjecture for integral Apollonian gaskets. That is, for a fixed integral, primitive Apollonian gasket, almost every (in the sense of density) admissible (passing local obstructions) integer is the curvature of some circle in the gasket.

64 pages, 3 figures; Appendix by Peter Varju

References in corpus (3)

• Growth in finite simple groups of Lie type of bounded rank
• Effective bisector estimate with application to Apollonian circle packings
• A proof of the positive density conjecture for integer Apollonian circle packings

Cited by in corpus (3)

• The Affine Sieve Beyond Expansion I: Thin Hypotenuses
• The local-global principle for integral bends in orthoplicial Apollonian sphere packings
• A Remark on Sieving in Biased Coin Convolutions