The fractal dimension of the 3D Apollonian packing is computed in [this paper](http://www.worldscientific.com/doi/abs/10.1142/S0218348X94000739). 


In the introduction, the authors cite three of Boyd's paper (Ref 2, 5, 6) to support that the fractal dimension (Hausdorff dimension of the complement) can be approximated by the growth rate of curvature (see Eq. (1)). This kind of estimates was also used for other space-filling packings, such as [this paper](http://www.worldscientific.com/doi/abs/10.1142/S0218348X04002549), also citing Boyd.


I may have missed something, but here is what I learned from the references:


* Ref 2 is talking about **exponent** of 3D osculatory packings.
* Ref 5 proved that fractal dimension equals exponent for **2D** Apollonian packing.
* Ref 6 proved that growth rate converges to the exponent for **2D** Apollonian packing. ($\limsup\to\lim$)


I do believe that the Hausdorff dimension of the complement equals the exponent of the packing, and they are approximated by the growth rate of curvatures. But I'm not convinced by the reference and the argument, especially for the "dimension=exponent" part.


I would appreciate if you can help to complete the chain of references.




---


**Edit:** I have now my own proof, which is very different from Boyd's approach. But I'm still not sure whether this is previously known. I add the label of hyperbolic geometry, in the hope that people from Kleinian group might know something.



For a 3D Apollonian packing, do we really know that the Hausdorff dimension of the complement is approximated by the growth rate of curvature?
OK, I found the reference:


For those who care, it's recently proved **in a much stronger form** by Oh and Shah in [The asymptotic distribution of circles in the orbits of Kleinian groups](http://www.ams.org/mathscinet-getitem?mr=2874933). The paper is about circle packing, but the method can be generalized to higher dimensions, as pointed out at the bottom of page 9.


