Arithmetic sums of cantor sets

It is easy to see that the Hausdorff dimension of a mid- $\left(1,-,2,\lambda \right)$ Cantor set is given by

Because Cantorvals have positive measure, they must have Hausdorff dimension 1.

Known results

A theorem from Mendes and Oliveira completely characterizes the types of topological structures that result from taking the sum of two homogeneous Cantor sets. In 1994 they proved the following result [link] .

Theorem 3.1 (Mendes and Oliveira 1994) If $C\left(\lambda ,,,A\right)$ and $C\left(\gamma ,,,B\right)$ are homogeneous Cantor sets, then we have one of the following possibilities:

• (i) $C\left(\lambda ,,,A\right)+C\left(\gamma ,,,B\right)$ is a Cantor set;
• (ii) $C\left(\lambda ,,,A\right)+C\left(\gamma ,,,B\right)=\left[0,,,2\right]$ ;
• (iii) $C\left(\lambda ,,,A\right)+C\left(\gamma ,,,B\right)$ is an L-Cantorval such that $\left[0,,,1\right]$ is contained in an interval of $C\left(\lambda ,,,A\right)+C\left(\gamma ,,,B\right)$ and 2 is a point of $C\left(\lambda ,,,A\right)+C\left(\gamma ,,,B\right)$ ;
• (iv) $C\left(\lambda ,,,A\right)+C\left(\gamma ,,,B\right)$ is an R-Cantorval such that $\left[1,,,2\right]$ is contained in an interval of $C\left(\lambda ,,,A\right)+C\left(\gamma ,,,B\right)$ and 0 is a point of $C\left(\lambda ,,,A\right)+C\left(\gamma ,,,B\right)$ ;
• (v) $C\left(\lambda ,,,A\right)+C\left(\gamma ,,,B\right)$ is an M-Cantorval such that 0 and 2 are points of $C\left(\lambda ,,,A\right)+C\left(\gamma ,,,B\right)$ .

In this study, the only possible structures for the sum of two mid- $\alpha$ Cantor sets are (i) , (ii) , and (v) . This is due to the fact that the sum of two symmetric sets must be symmetric.

Another well-known result gives a sufficient, but not necessary, condition for the sum of two mid- $\alpha$ Cantor sets being another Cantor set, though not necessarily homogeneous. This result is presented in the following Lemma.

Lemma 3.2 For any two sets $K_1,K_2\subseteq \mathbb{R}$ ,

[Sketch of Proof] We offer a quick sketch of the proof in the case where $K_1$ and $K_2$ are mid- $\alpha$ Cantor sets, based on a method described in [link] . Let $\pi$ be the projection of ${\mathbb{R}}^2$ to the x-axis in the direction of the vector $\left(1,,,-,1\right)$ . Then,

See Figures [link] and [link] . It is easy to see from the definition of Hausdorff dimension that

It can also be seen from the figures that

This finishes the sketch of the proof of Lemma 3.2.

This result tells us that if $HD\left(C,\left(\lambda \right)\right)+HD\left(C,\left({\lambda }^{\theta }\right)\right)<1$ , then $C\left(\lambda \right)+C\left({\lambda }^{\theta }\right)$ must be a Cantor set with zero Lebesgue measure. This area can be seen below in [link] as the region below the green line, given by $\lambda =2^{-\left(1,+,1,/,\theta \right)}$ .

As mentioned earlier, this condition is not necessary. It is known that for rational $\theta$ , there are $\lambda$ such that $HD\left(C,\left(\lambda \right)\right)+HD\left(C,\left({\lambda }^{\theta }\right)\right)>1$ but $C\left(\lambda \right)+C\left({\lambda }^{\theta }\right)$ is a Cantor set with zero Lebesgue measure. We will have more to say about this when we discuss our new results in "Results from the Study" .

A partial converse to this was shown for irrational $\theta$ by Peres and Shmerkin in [link] .

Theorem 3.3 (Peres and Shmerkin 2009) Let $C\left(\lambda \right)$ and $C\left({\lambda }^{\theta }\right)$ be two mid- $\alpha$ Cantor sets. Then, if $\theta$ is irrational,

There are also results pertaining to sufficient conditions for when the sum of two mid- $\alpha$ Cantor sets is the full interval $\left[0,,,2\right]$ . The following result is also from Mendes and Oliveira [link] .

Theorem 3.4 (Mendes and Oliveira 1994) Let $C\left(\lambda \right)$ and $C\left({\lambda }^{\theta }\right)$ be two mid- $\alpha$ Cantor sets, with $\theta \ge 1$ . For each $\theta \ge 1$ , we denote by ${\lambda }_1\left(\theta \right)\in \left(0,,,\frac{1}{2}\right)$ the unique solution to the equation ${\lambda }^{\theta }=1-2\lambda$ , and we denote by ${\lambda }_2\left(\theta \right)\in \left(0,,,\frac{1}{2}\right)$ the unique solution to $\frac{\lambda }{1-\lambda }+\frac{{\lambda }^{\theta }}{1-{\lambda }^{\theta }}$ . Note that ${\lambda }_1\left(1\right)={\lambda }_2\left(1\right)=\frac{1}{3}$ and ${\lambda }_1\left(\theta \right)<{\lambda }_2\left(\theta \right)$ for all $\theta >1$ . With this notation, we have the following: