# Apollonian gasket and the degree of convergence

Let $r_1,r_2\dots$ be the radii of Apollonian gasket. I would like to know for which values $\alpha$ we have $$\sum_{n=1}^\infty r_n^\alpha<\infty.$$

I know that if three circles $A$, $B$ and $C$ are tangent to two circles $D_1$ and $D_2$ then $$d_1+d_2=2(a+b+c),$$ where $a$, $b$, $c$, $d_1$ and $d_2$ denote the curvatures of the corresponding circle. (For example, on the picture, $3+3=2(2+2-1)$.)

In principle, it gives a recursive formula for $r_n$, but I was not able to figure out how to use it.

Motivation: I would like to know if one can cover whole measure of a square with countable number of disjoint open discs with radii $r_1,r_2\dots$ such that $$\sum_{n=1}^\infty r_n^\alpha<\infty$$ for all $\alpha>1$. By some reason I believe that Apollonian gasket is optimal in this sense; at least it worth to check it.

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For what its worth, while in principle one has a recursive formula for the curvatures, you are not alone in being unable to effectively use it. My advisor in grad school spent some time looking at the question of 'given starting curvatures of a gasket what curvatures can/cannot appear in the gasket?' without doing an exhaustive depth search. As far as I know this question is still open (the best one can do is find some modular restrictions), so short of finding a better answer, you probably wouldn't be able to calculate the critical $\alpha$ without a computer like in Gerald's answer. – ARupinski Jan 17 '13 at 3:56

This critical value is $\alpha_0 \approx 1.3056867$ ... For $\alpha > \alpha_0$ the series converges, for $\alpha < \alpha_0$ it diverges.

Before the inexpensive computer, it was difficult to tell whether the critical value is ${}> 1$ or not.

Boyd, David W. The sequence of radii of the Apollonian packing. Math. Comp. 39 (1982), no. 159, 249–254.

http://www.ams.org/mathscinet-getitem?mr=658230

mentioned in the comments... arbitrarily packed disks, not necessarily touching as in Apollonian packings. The critical value (= dimension of the residual set) is shown to be ${}> 1.02$.
You get the same critical value with different starting curvatures. Maybe it is the 1.02 I was thinking of when I said it was hard to tell if it was ${}>1$. – Gerald Edgar Jan 17 '13 at 14:58