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with the following constraints on $A$ :
As in the case of the mid- $\alpha $ Cantor sets $C\left(\lambda \right)$ , homogeneous Cantor sets exhibit self-similarity in the following sense:
An example of this self-similarity can be seen in [link] .
Another relevant topological structure is the Cantorval . In loose terms, one could consider Cantorvals as "Cantor sets that contain intervals." To be more precise about this definition, we need to first define a gap of a set to be a bounded connected component of the complement. For example, in the Cantor ternary set $T$ , the interval $\left(\frac{1}{3},,,\frac{2}{3}\right)$ is the largest gap of $T$ .
Now we can formally define a Cantorval. We say that a compact, perfect set $C\subseteq \mathbb{R}$ is an M -Cantorval if every gap of $C$ is accumulated on each side by other gaps and intervals of $C$ . An example of an M -Cantorval is given by Anisca and Chlebovec in [link] ; see [link] .
Similarly, we say that $C$ is an L -Cantorval (or an R -Cantorval ) if every gap of $C$ is accumulated on the left (or right) by gaps and intervals of $C$ , and if each gap of $C$ has an interval adjacent to its right (or left). See [link] .
The problem tackled in this study revolves around characterizing the topological properties of the sum of two mid- $\alpha $ Cantor sets $C\left(\lambda \right)$ and $C\left(\gamma \right)$ , given by
in terms of $\lambda $ and $\gamma $ .
It is known that this sum can be an interval, as in the case of $C\left(\frac{1}{3}\right)+C\left(\frac{1}{3}\right)=\left[0,,,2\right]$ . However, such a sum can result in another Cantor set, as with $C\left(\frac{1}{5}\right)+C\left(\frac{1}{5}\right)$ . The proofs of these facts are in "Known Results" .
When studying this sum, it is more convenient to characterize it in terms of $\lambda $ and ${\lambda}^{\theta}=\gamma $ with $\theta \ge 1$ as opposed to simply just $\lambda $ and $\gamma $ . This is due to a result from [link] discussed below.
A useful way of characterizing these types of sum sets in terms of Hausdorff dimension . To define Hausdorff dimension, as done in [link] , we need first to define the Hausdorff $\alpha $ -measure . (Note that the $\alpha $ here is different from the $\alpha $ used to define the mid- $\alpha $ Cantor sets.)
Given a set $K\subseteq \mathbb{R}$ and a finite covering $\mathcal{U}={\left\{{U}_{i}\right\}}_{i\in I}$ of $K$ by open intervals, we define ${\ell}_{i}$ to be the length of ${U}_{i}$ , and then $\text{diam}\left(\mathcal{U}\right)$ to be the maximum of the ${\ell}_{i}$ . Then, the Hausdorff $\alpha $ -measure ${m}_{\alpha}\left(K\right)$ of $K$ is
Then, there is a unique number $HD\left(K\right)$ such that for $\alpha <HD\left(K\right)$ , ${m}_{\alpha}\left(K\right)=\infty $ , and for $\alpha >HD\left(K\right)$ , ${m}_{\alpha}\left(K\right)=0$ . We call this number the Hausdorff dimension of $K$ .
From this definition, it is clear that any set with Hausdorff dimension less than 1 must have zero Lebesgue measure. Note that it is possible to have a Cantor set that has both Hausdorff dimension equal to 1 and Lebesgue measure zero. There is also another class of Cantor sets that have Hausdorff dimension 1 and positive Lebesgue measure.
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