Is there any sense in which the "epsilon-tail" of an Apollonian disk-packing (by which I mean the union of the disks of radius less than epsilon) exhibits more and more statistical isotropy as epsilon goes to zero?
Here's one example of the kind of thing I mean. Label the disks $D_1,D_2,\dots$ in order of weakly decreasing radius. It seems likely that for all $n$ in a set of density 1, there is a unique largest disk $D'_n$ tangent to disk $D_n$, and the ray from the center of $D_n$ to the center of $D'_n$ determines a unit vector $v_n$. Now we can ask whether the end-points of the unit vectors $v_1,v_2,\dots$ are uniformly distributed on the circle.
I'm interested in finding out what's known about questions like this.