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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).

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  • $\begingroup$ Isn't the question of whether $\sqrt2$ normal unknown? (Or you think you're about to solve it?) $\endgroup$ – Akiva Weinberger May 18 '15 at 12:09
  • $\begingroup$ "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. $\endgroup$ – Henning Makholm May 18 '15 at 12:18
  • $\begingroup$ @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). $\endgroup$ – A.P. May 18 '15 at 12:23
  • $\begingroup$ @HenningMakholm Thanks for the remark. Sadly, it makes that result slightly less interesting, making this question even more relevant for me. $\endgroup$ – A.P. May 18 '15 at 12:25
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The NSA uses the square roots of small primes in the SHA1 and SHA2 algorithms. https://en.wikipedia.org/wiki/Nothing_up_my_sleeve_number These are referred to as nothing up my sleeve numbers.

In cryptography, nothing up my sleeve numbers are any numbers which, by their construction, are above suspicion of hidden properties. They are used in creating cryptographic functions such as hashes and ciphers. These algorithms often need randomized constants for mixing or initialization purposes. The cryptographer may wish to pick these values in a way that demonstrates the constants were not selected for (in Bruce Schneier's words) a “nefarious purpose”, for example, to create a “backdoor” to the algorithm. These fears can be allayed by using numbers created in a way that leaves little room for adjustment. An example would be the use of initial digits from the number π as the constants. Using digits of π millions of places into its definition would not be considered as trustworthy because the algorithm designer might have selected that starting point because it created a secret weakness the designer could later exploit.

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    $\begingroup$ I don't understand. How is this an application of normal numbers? $\endgroup$ – A.P. Sep 10 '15 at 13:41
  • $\begingroup$ Random number generators are used in cryptography. Random number generators sometimes use normal numbers. See Random Number Generators And Normal Numbers: emis.de/journals/EM/expmath/volumes/11/11.4/pp527_546.pdf $\endgroup$ – Russell Harkins Sep 10 '15 at 14:25

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