The sequence of radii of the Apollonian packing

Abstract

We consider the distribution function N ( x ) N(x) of the curvatures of the disks in the Apollonian packing of a curvilinear triangle. That is, N ( x ) N(x) counts the number of disks in the packing whose curvatures do not exceed x. We show that log ⁡ N ( x ) / log ⁡ x \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 y = A{n^s} to the values of N − ( 1000 n ) {N^ - }(1000n) for n = 1 , 2 , … , 6400 n = 1,2, \ldots ,6400 produces the estimate S ≈ 1.305636 S \approx 1.305636 which is consistent with the known bounds 1.300197 > S > 1.314534 1.300197 > S > 1.314534 .