Counting problems in Apollonian packings

Abstract

Abstract. An Apollonian circle packing is a classical construction which is made by repeatedly inscribing circles into the triangular interstices in a Descartes configuration of four mutually tangent circles. Remarkably, if the original four circles have integer curvature, all of the circles in the packing will have integer curvature, making the packings of interest from a number theoretic point of view. Many of the natural arithmetic problems have required new and sophisticated tools to solve them. The reason for this difficulty is that the study of Apollonian packings reduces to the study of a subgroup of GL 4 (Z) that is thin in a sense that we describe in this article, and arithmetic problems involving thin groups have only recently become approachable in broad generality. In this article, we report on what is currently known about Apollonian packings in which all circles have integer curvature and how these results are obtained. This survey is also meant to illustrate how to treat arithmetic problems related to other thin groups.

The Apollonian structure of Bianchi groups

“…The study of Descartes quadruples has a lively history and there are several excellent expositions; see for example [10,31]. If one begins with three mutually tangent circles with curvatues Adding these two new circles to our original triple, we have a set of 5 circles.…”

“…Significant progress has been made toward this conjecture, most notably that it holds for a set of integers of density one [3] (positive density was first shown in [2]). For an excellent overview and further references, see [10]; see also the series of papers [12,13,14] which are central to the field, and the exposition [31]. For the related question of the multi-set of integral curvatures appearing in a packing, a gateway to the literature is the survey [29].…”

Descartes Circle Theorem, Steiner Porism, and Spherical Designs

“…The Descartes Circle Theorem has been popular lately because it underpins the geometry and arithmetic of Apollonian packings, a subject of great current interest; see, e.g., surveys [7,8,9,13]. In this article we revisit this classic result, along with another old theorem on circle packing, the Steiner porism, and relate these topics to spherical designs.…”

Generic Thinness in Finitely Generated Subgroups of SL$_n(\mathbb Z)$

“…This question is motivated in part by recent developments in number theory ( [4], [22], [23], etc.) which have made approachable previously unsolved arithmetic problems involving thin groups (see, for example, [10] and [18] for an overview). Given these new ways to handle such groups in arithmetic settings, it has become of great interest to develop a better understanding of thin groups in their own right: for example, [11], [25], and [5] have answered the question of telling whether a given finitely generated group is thin (given in terms of its generators) in various settings.…”