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!

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    $\begingroup$ Much is known about sums of affine self-similar Cantor sets. However, I suspect that in the case of Fuchsian limit sets you are on your own. You may want to look at the methods by which the affine self-similar case is analyzed and try to use them in your setting. $\endgroup$ – Misha Oct 4 at 23:12
  • $\begingroup$ Dear Misha, thank you for your reply. I looked at the methods by which the affine self-similar case is analyzed, but it does not seem to work on my setting. Do you know any method to approximate a limit set of Fuchsian group by a sequence of affine self-similar Cantor sets? $\endgroup$ – DyNaMic Oct 8 at 13:04

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