Statistical regularity of Apollonian gaskets

Abstract

Apollonian gaskets are formed by repeatedly filling the gaps between three mutually tangent circles with further tangent circles. In this paper we give explicit formulas for the limiting pair correlation and the limiting nearest neighbor spacing of centers of circles from a fixed Apollonian gasket. These are corollaries of the convergence of moments that we prove. The input from ergodic theory is an extension of Mohammadi–Oh’s theorem on the equidisribution of expanding horospheres in the frame fundles of infinite volume hyperbolic spaces.