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

Integral Apollonian circle packing defined by circle curvatures of (−1, 2, 2, 3) http://upload.wikimedia.org/wikipedia/commons/thumb/7/7a/ApollonianGasket-1_2_2_3-Labels.png/480px-ApollonianGasket-1_2_2_3-Labels.png

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.

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.