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Yesterday I had a discussion about Conway's base-13 (https://en.wikipedia.org/wiki/Conway_base_13_function) function (and what fancy properties it has). During that discussion the other person explained that he pictured the level sets to look like a fractal. This got me curious and motivated the following question:

Is it known what the Hausdorff dimension of the level sets of Conway's base-13 function is?

In case you might worry that it is not measurable (it is) see Is Conway's base-13 function measurable?

Severin Schraven
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  • Surely there are countably many preimages to each non-zero real, so that $\text{dim}_H(C^{-1}(\alpha))=0$ for any $\alpha\ne 0$. On the other hand, $C^{-1}(0)$ is a set of measure 1, so has Hausdorff dimension 1. – Anthony Quas May 09 '19 at 23:48
  • @AnthonyQuas Thank you for the answer. The second point was clear to me, but I completely missed the the first one. – Severin Schraven May 10 '19 at 06:48

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