On the Local-Global Conjecture for integral Apollonian gaskets
  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. 
Jean Bourgain and Alex Kontorovich
math.NT