Background: I am writing a master thesis on the complexity of the expansions of algebraic numbers in some complex basis $\beta$ with $|\beta| > 1$. This is a very small step towards proving the
Conjecture. Every irrational algebraic number is absolutely normal.
Here a real number $\alpha$ is said to be normal in base $b$ or $b$-normal (where $b > 1$ is an integer) if every sequence of $n$ consecutive digits (for every positive integer $n$) appears in the $b$-ary expansion of $\alpha$ with the same frequency. Further, a number is said to be absolutely normal if it is normal in every integer base $b > 1$.
Question: This led me to wonder if there is any known application for normal numbers, i.e. some (proved or conjectured) statement of the form
If $\alpha$ is normal, then [something interesting in terms of $\alpha$] happens.
I'm asking here because, while I could find a wealth of information regarding normal numbers per se, I couldn't find any uses for them. The only interesting result I was able to find is that normal sequences cannot be compressed by a lossless finite-state compressor. In other words (if I interpret this correctly), there is no way to encode the $b$-ary expansion of a normal number with a shorter sequence of finitely many less symbols (without losing information).