# Arithmetic sums of cantor sets

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$C\left(\lambda ,,,A\right)=\sum _{n=0}^{\infty }{\alpha }_{n}{\lambda }^{n}\phantom{\rule{1.em}{0ex}}\text{where}\phantom{\rule{1.em}{0ex}}{\alpha }_{n}\in A\phantom{\rule{3.33333pt}{0ex}}\forall n$

with the following constraints on $A$ :

${a}_{0}=0,\phantom{\rule{1.em}{0ex}}{a}_{k}=1-\lambda ,\phantom{\rule{1.em}{0ex}}\left({a}_{i+1},-,{a}_{i}\right)>\lambda \phantom{\rule{3.33333pt}{0ex}}\text{for}\phantom{\rule{3.33333pt}{0ex}}0\le i\le k-1.$

As in the case of the mid- $\alpha$ Cantor sets $C\left(\lambda \right)$ , homogeneous Cantor sets exhibit self-similarity in the following sense:

$C\left(\lambda ,,,A\right)=\bigcup _{j=0}^{k}\left(\lambda ,·,C,\left(\lambda ,,,A\right),+,{a}_{j}\right).$

An example of this self-similarity can be seen in [link] . The first, second, and third stages in the construction of a homogeneous Cantor set with λ = 0 . 2 and A = 0 , 0 . 3 , 0 . 8 .

## Cantorvals

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] . An M -Cantorval constructed in a manner similar to the construction of the Cantor ternary set T , except one only removes intervals at the odd stages; no intervals get removed at even stages. An example of an L -Cantorval. Note (or, given the limited resolution, imagine) that every gap has an interval on its right and is accumulated on the left by points, intervals, and gaps.

## Sums of mid- $\alpha$ Cantor sets

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

$C\left(\lambda \right)+C\left(\gamma \right)=\left\{x,+,y,\mid ,x,\in ,C,\left(\lambda \right),,,\phantom{\rule{3.33333pt}{0ex}},y,\in ,C,\left(\gamma \right)\right\}$

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.

## Hausdorff dimension

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

${m}_{\alpha }\left(K\right)=\underset{\epsilon \to 0}{lim}\left(\underset{\begin{array}{c}\mathcal{U}\phantom{\rule{0.166667em}{0ex}}\text{covers}\phantom{\rule{0.166667em}{0ex}}K\\ \text{diam}\left(\mathcal{U}\right)<\epsilon \end{array}}{inf},\left(\sum _{i\in I},{\ell }_{i}^{\alpha }\right)\right).$

Then, there is a unique number $HD\left(K\right)$ such that for $\alpha , ${m}_{\alpha }\left(K\right)=\infty$ , 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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