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 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. arXiv:1709.00676v2 [math.DS]

Directions in orbits of geometrically finite hyperbolic subgroups

“…Herein we will answer these questions and give a full characterisation of the spatial statistics of such a point set as viewed from a fixed observer in hyperbolic space or its boundary. These questions have been addressed previously for lattices [1,7,13,20], and for certain thin groups [26,27]. However we will treat a much more general class of subgroups in arbitrary dimension.…”

“…Zhang then proved a limiting theorem for the gap distribution of directions for certain Schottky groups [26] (hence this was the first treatment of the infinite volume case, in 2 dimensions). Following this, Zhang proved a limiting distribution for the directions of centers of Apollonian circle packings [27] (another nonlattice example, this time in 3 dimensions). As an application of one of our main theorems (Theorem 4•2), in Subsection 2•4 we will discuss how our methods apply to a general class of sphere packings.…”

“…Asymptotic counting formula for these packings are given in [16, theorem 7.5]. And, in the Apollonian case for n = 3, [27] studied the spatial statistics of the centers of these packings. In fact, a special case of Theorem 4•2 (below) characterizes the spatial statisics of these packings.…”