Isotropy of Apollonian disk-packing

Question

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.

Answer 1

Hee Oh has replied to my question and given me permission to post her reply here. What follows are her words (so "I" means "Hee") with my reformatting.

What I proved with Shah (Inventiones, 2012) says that for a given Apollonian circle packing $P$, considered as a countable union of circles in the plane, "small circles in $P$ are uniformly distributed with respect to the $\alpha$-dimensional Hausdorff measure of the residual set of $P$, which is the closure of $P$", where $\alpha=1.305...$. For any region $E$ with piecewise smooth boundary the number of circles intersecting $E$ of radius at least $t$ is asymptotic to $c t^{-\alpha} H_{\alpha}(E)$.

Chapter 8 of the following article can be useful: http://gauss.math.yale.edu/~ho2/newMSRI_Oh.pdf

So it seems that your question amounts to asking how the $\alpha$-dimensional Haudorff measure of the residual set of $P$ (${\rm Res}(P$)) behaves. Since $\alpha$ is bigger than 1, a theorem of Marstrand says that a typical point $x$ in ${\rm Res}(P$) -- typical in the sense of the $\alpha$-dimensional Hausdorff measure -- is a condensation point for ${\rm Res}(P$), meaning that for almost all directions $\theta$, $x$ is a limit point of the intersection of $(x, \theta)$ and Res($P$) where $(x, \theta)$ is the ray from $x$ in the direction of $\theta$.

I am referring to Theorem 6.1 of the following paper http://gauss.math.yale.edu/~ho2/BR_JAMS_Final.pdf where we copied a theorem of Marstrand in 1954.

So, you do see circles in almost all directions in the above sense.

Link to Marstrand's paper: http://dx.doi.org/10.1112/plms/s3-4.1.257