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The fractal dimension of the 3D Apollonian packing is computed in this paper.

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, 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.

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  • $\begingroup$ The dimension=exponent part should be known from some sort of Patterson-Sullivan theory, as some integrability condition for proper Poincare series, especially if you're interested in only rank $1$ spaces. I would check Oh's papers, especially the papers with Mohammadi (If I recall correctly the first counting result was achieved with Kontorovich, and the second one with Mohammadi). McMullen have done some work computing dimensions as well (late 90's or so). $\endgroup$
    – Asaf
    Commented Sep 14, 2014 at 10:02
  • $\begingroup$ @Asaf, what do you mean by rank 1 spaces? Do you mean pseudo-Euclidean spaces of signature (1,n)? I am indeed interested to know more about any sort of P-S theory in spaces of signature (p,q). $\endgroup$
    – Hao Chen
    Commented Sep 16, 2014 at 15:57
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    $\begingroup$ one of the definitions of a rank of symmetric space is the rank of the group from which the symmetric space is formed, so indeed, SO(1,n) are rank $1$. For higher rank, Quint proved analogues of PS for Schottky groups in his PhD thesis (which are doable due to the symbolic coding), and I think some results in the general case were proven by Albuquerque. $\endgroup$
    – Asaf
    Commented Sep 16, 2014 at 16:25

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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. The paper is about circle packing, but the method can be generalized to higher dimensions, as pointed out at the bottom of page 9.

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