# Is there any known application for normal numbers?

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).

• Isn't the question of whether $\sqrt2$ normal unknown? (Or you think you're about to solve it?) – Akiva Weinberger May 18 '15 at 12:09
• "Finite-state compressor" is a quite strong assumption on the compressor. If you relax the finite-state requirement, every computable normal number (such as the Champernowne constant) can be "compressed" to the program that computes it. – hmakholm left over Monica May 18 '15 at 12:18
• @columbus8myhw Of course not! That's why I said that it is a very small step. :) I extended a proof of Adamczewski and Bugeaud where they showed that the complexity function (i.e. the number of different blocks of a given length) of the $b$-ary expansion of an irrational algebraic number grows more than linearly. Note that the $b$-ary expansion of every $b$-normal number has exponential growth (though the converse isn't true). – A.P. May 18 '15 at 12:23
• @HenningMakholm Thanks for the remark. Sadly, it makes that result slightly less interesting, making this question even more relevant for me. – A.P. May 18 '15 at 12:25