The Hausdorff, packing, and Minkowski dimensions of general Cantor-type sets

Question

I want to find a reference for the proof of the following statements:

If $C$ is a Cantor middle $\{\alpha_n\}$-set, $0\leq \alpha_n\leq 1$, $a_n:=\prod_{j=1}^n(1-\alpha_j)/2$, then:

1. $\dim_P(C)={\mkern 1.5mu\overline{\mkern-1.5mu\dim_\mkern-1.5mu}\mkern 1.5mu}_M(C)=\limsup_{n\to \infty}n\log 2/(-\log a_n)$.
2. $\dim_H(C)=\underline{\dim}_M(C)=\liminf_{n\to \infty}n\log 2/(-\log a_n)$.

Here, $\{\alpha_n\}$ is the proportion of intervals removed from the middle in the $n$-th step. For example, for the standard Cantor middle third set, $\alpha_n=1/3$ for all $n$.

$\dim_P$ is the packing dimension.

Answer 1

I believe that the result you want in your question follows from putting together several pieces of the literature on fractal sets, along with a little bit of work.

First, I believe the equality of the upper and lower Minkowski dimensions to the quantities you write is a Corollary of Proposition 3.6 of "Techniques in Fractal Geometry" by Falconer. This proposition gives a formula for the upper and lower Minkowski definitions of sets defined by cutting out intervals out of some base interval. After the Proposition, Falconer describes how to use this result to compute the Minkowski dimension of the middle third's Cantor set, and similar reasoning will deduce the formulas in your question.

I am not as certain on how to deduce the equality of the Hausdorff and packing dimensions to these quantities. Certainly, one inequality is automatic for each: the lower Minkowski dimension always is an upper bound for the Hausdorff dimension and the upper Minkowski dimension is always an upper bound for the packing dimension.

To equate the packing and upper Minkowski dimensions, I suggest looking at Section 2.7 and 2.8 of "Fractals in Probability and Analysis" by Bishop and Peres, paying particular attention to Corollary 2.8.2, which gives a sufficient condition for the equality of the packing and upper Minkowski dimension. In this situation, the Cantor set structure of your set is likely essential and needs to be used in your proof.

To equate the lower Minkowski dimension and Hausdorff dimension, I would likely attempt to do this directly by constructing a suitable measure on the Cantor set and using the Mass Distribution Principle: see Lemma 1.2.8 of the book by Bishop and Peres for the statement and surrounding discussion. Again, to construct the measure, you will likely want to directly exploit the Cantor set structure. I haven't worked out the details but I am optimistic this will work - the Mass distribution principle is one of our best and most versatile tools for lower bounding Hausdorff dimension.

I hope this is helpful - I know the question has been closed open for a while now. While I apologize for the lack of precise statements, all the literature I discussed can be found for free via a quick internet search.