A Proof of the Positive Density Conjecture for Integer Apollonian Circle Packings

In the first picture in Figure 1 there are three mutually tangent circles packed in a large circle on the outside, with four curvilinear triangles in-between. Such a configuration of circles is called a Descartes configuration and is the starting point for a bounded Apollonian circle packing (ACP). One may also consider an unbounded Apollonian circle packing in which the original Descartes configuration consists of four circles which are all externally tangent to each other, or where one or two of the circles are in fact straight lines. One such configuration is depicted in the first picture of Figure 2; the two parallel lines can be thought of as circles of infinite radius tangent at infinity. The packing in Figure 2 is the only kind of unbounded packing relevant to this paper. Once a Descartes configuration is given, one constructs an ACP by inscribing a circle into each of the four triangular interstices in the original Descartes configuration, as in the second picture of both Figure 1 and Figure 2. This results in 12 new triangular interstices which are in turn filled with circles. An ACP is then a packing of infinitely many circles obtained by continuing this process indefinitely. We note that this construction is well defined once the original four circles are given. An old theorem (circa 200 BC) of Apollonius of Perga states that there are precisely two circles tangent to all of the circles in a triple of mutually tangent circles or lines. It follows that each triangular interstice arising in the construction above can be packed with precisely one circle.