The sequence of radii of the Apollonian packing

Abstract:

We consider the distribution function $N(x)$ of the curvatures of the disks in the Apollonian packing of a curvilinear triangle. That is, $N(x)$ counts the number of disks in the packing whose curvatures do not exceed x. We show that $\log N(x)/\log x$ approaches the limit S as x tends to infinity, where S is the exponent of the packing. A numerical fit of a curve of the form $y = A{n^s}$ to the values of ${N^ - }(1000n)$ for $n = 1,2, \ldots ,6400$ produces the estimate $S \approx 1.305636$ which is consistent with the known bounds $1.300197 < S < 1.314534$.