Let $L \subset \mathbb{R}$ be a limit set of a Fuchsian group $\Gamma$. If $\Gamma$ is a non-elementary Fuchsian group of second kind, then $L$ is a Cantor set. For example: $\Gamma= \bigg\langle \begin{bmatrix}1&0\\3&1\end{bmatrix},\begin{bmatrix}13/5&12/5\\12/5&13/5\end{bmatrix} \bigg\rangle$.

What is known about the arithmetics on $L$ and itself i.e., the set $L*L:=\{x*y|x,y\in $L$\}$, where $*\in \{+,-,\times,\div\}$?

There are many results on the general Cantor set, with sufficient conditions on the thickness of the Cantor set so that the resulting set is an interval. But it seems like there is no research on the arithmetics on the limit sets in particular, for instance, the example above.

I hope someone can give me an idea or any reference to this question. Thank you in advance!