Statistical Regularity of Apollonian Gaskets

TL;DR

This paper derives explicit formulas for the statistical distribution of circle centers in Apollonian gaskets, revealing their regularity through advanced ergodic theory and moment convergence analysis.

Contribution

It provides the first explicit formulas for the pair correlation and nearest neighbor spacing in Apollonian gaskets, extending ergodic theory applications.

Findings

Explicit formulas for pair correlation of circle centers

Explicit formulas for nearest neighbor spacing

Convergence of moments established

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 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 infinite volume hyperbolic spaces.