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It's easy to construct an example of a number that's normal in a given base, but for most given numbers it's notoriously hard to prove that they're normal.

Has any number ever been proven to be normal (either in a particular base or in all bases) that wasn't specifically constructed to be normal? Say, a number that had already been defined and explored in a previous paper not related to normality, but that was only proven to be normal in a later paper. (An answer to this question doesn't need to actually provide the two papers; I'm just explaining what I'm looking for. It doesn't count if the original paper contained an "almost proof" of normality, and then a later paper just filled in a few missing steps.)

The only possible example that I know of are the Chaitin's constants, although I'm not sure whether it was immediately realized that they are normal, or whether it was simple proof once someone thought to ask the question. (Also, Chaitin’s constants are not computable, while I’d prefer an example that’s a computable number.) Ideally, I'd like an example of a highly nontrivial proof of normality for a well-studied number, such that the finding of normality was very "surprising".

I'd also be interested in "nontrivial" proofs of the non-normality of a previously studied number, but I admit that in this case it's hard to pin down exactly what I mean by "nontrivial", because (e.g.) every rational number is not normal.

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    $\begingroup$ Maybe .2357111317192329... (concatenated prime numbers in order)? $\endgroup$
    – Noam D. Elkies
    Commented Dec 22, 2022 at 2:04
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    $\begingroup$ @tparker It was not proven immediately upon its first introduction, because it was conjectured by Champernowne to be normal, but not proved to be normal until the Copeland–Erdős paper. But Champernowne's paper was all about normality, so it doesn't provide an example of what you're looking for. $\endgroup$
    – Timothy Chow
    Commented Dec 22, 2022 at 12:55
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    $\begingroup$ As for the area of math, according to the Mathematics Subject Classification, it's 11K16, where 11 is number theory, and 11K is "Probabilistic theory: distribution modulo 1; metric theory of algorithms." (By the way, I was surprised to discover that the AMS website for the MSC is now paywalled behind MathSciNet, even though the MSC is covered by a Creative Commons CC-BY-NC-SA license. Fortunately, zbMATH Open still makes the MSC freely available. $\endgroup$
    – Timothy Chow
    Commented Dec 22, 2022 at 13:06
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    $\begingroup$ @TimothyChow It is visible to me. mathscinet.ams.org/mathscinet/freeTools.html?version=2 $\endgroup$
    – Brendan McKay
    Commented Dec 22, 2022 at 14:19
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    $\begingroup$ @BrendanMcKay Ah...it seems that my bookmark was outdated. I retract my claim. $\endgroup$
    – Timothy Chow
    Commented Dec 22, 2022 at 14:23

1 Answer 1

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Quoting from the 2018 Normal numbers and computer science, by V.Becher and O.Carton (In Sequences, groups, and number theory (pp. 233-269). Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-69152-7_7):

"All known examples of normal numbers have been obtained by constructions."

This is not too surprising, if true. This claim is not completely clear (see also the comments below), however it could be somewhat weaker than "have been constructed specifically to show an example of normality". For instance, the Champernowne constant is basic in symbolic dynamics (the expansion of its base-2 version is the model-case of a transitive point, and a recurrent but not uniformly recurrent point, under the shift), and I don't think it was defined by Champernowne in his 1933 paper just, or primarily, to show an example of normal number (but see Timothy Chow's comment below).

This example answers also a question you posed in a comment: normal numbers have important connections with the theory of dynamical systems, among other things. See for instance the 2006 reference work Old and new results on normality, by M.Queffélec.

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    $\begingroup$ While it's possible that the "Champernowne constant" predates Champernowne, his paper cites no previous literature and gives no indication that he is interested in anything other than normality. $\endgroup$
    – Timothy Chow
    Commented Dec 22, 2022 at 12:50
  • $\begingroup$ Oh well, so it confirms the claim. $\endgroup$
    – Alessandro Della Corte
    Commented Dec 22, 2022 at 12:51
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    $\begingroup$ Isn't “have been obtained by constructions” a completely vacuous statement? Everything in maths is constructed in a sense. What's actually interesting is whether they have been constructed as a list of digits, or in some way that doesn't involve any base expansion. $\endgroup$
    – leftaroundabout
    Commented Dec 22, 2022 at 22:46
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    $\begingroup$ Chaitin's constant, the probability that a randomly constructed program will halt, does not look to me as defined "by constructions involving explicitly the digits of its expansion in some base". $\endgroup$
    – Bogdan Grechuk
    Commented Dec 22, 2022 at 23:24
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    $\begingroup$ I think that "obtained by constructions" is intended to mean "constructed specifically for the purpose of proving their normality." But I agree that Chaitin's constant looks like a counterexample to that version of the claim. $\endgroup$
    – Timothy Chow
    Commented Dec 23, 2022 at 4:14

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