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I am working a bit with problems in non-archimedean settings inspired by the famous periods conjecture by Kontsevich-Zagier. I was preparing a talk and wanted to give of background about the initial motivation, namely periods. A period in my definition is a complex number whose real and imaginary parts are both values of an integral over a subset of $\mathbb{R}^n$ defined by inequalities of polynomials over $\mathbb{Q}$, where the integrand can be an arbitrary rational function with rational coefficients (or equivalently actually just 1).

I know that there are a lot of open problems around periods, besides the famous conjecture itself it is not known whether e.g. $e$ is a period or not, whether $\frac{1}{\pi}$ is a period or not etc. What I could not find is an example of a real number that is known not to be a period and might it just be by some obscure construction as in the case of say normal numbers. I usually rather work with $p$-adic integrals, so I also don't know how such an construction would look like, so my question is summarized as:

Do we know a specific computable real number, that is provably not a period and if so, how do we construct it?

Florian Felix
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  • Euler-Masceroni constant is likely not a period. – Anixx Feb 07 '23 at 10:10
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    I mean almost any given real number is likely not a period, right? I would like to know whether we know for any real number whether it is not one. – Florian Felix Feb 07 '23 at 10:15
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    @Anixx But we do not even know whether the Euler-Mascheroni constant is irrational. If I understand the question right, it is about KNOWN such numbers. – Peter Feb 07 '23 at 10:23
  • Also, I am confused about the definition : "polynomial inequalities" ? And which subsets are allowed ? Surely not an arbitary interval , since then we could of course find an intergal with the given value. – Peter Feb 07 '23 at 10:27
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    The domain and the integrand must only involve rational fractions with rational coefficients. So the set of periods is countable, and you can use Cantor's diagonal argument to construct a non-period. – Anne Bauval Feb 07 '23 at 10:32
  • @AnneBauval I expected that, but this is not what was written down. Also, the question is surely to give a CONCRETE example of a known such number. – Peter Feb 07 '23 at 10:33
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    Reading the wiki, it seems that apart non-calculable numbers which are by definition not periods (e.g. Chaitin constant), we do not currently know if any of our beloved constants suspected not to be period effectively are. However it also cites this paper to construct calculable artificial ones, i.e. "non elementary numbers" as periods are "elementary" (elementary being a number that fulfil some Liouville criteria). – zwim Feb 07 '23 at 10:38
  • @AnneBauval But it answers the part whether "almost every" real number is real. – Peter Feb 07 '23 at 10:38
  • Florian you should edit your post (and well prepare your talk) to give a correct definition of the periods (the domain and the integrand must only involve rational fractions with rational coefficients). And doesn't your "might it just be by some obscure construction as in the case of say normal numbers" mean the CONTRARY of @Peter 's "the question is surely to give a CONCRETE example of a known such number"? I mean: do you accept "artificial" examples? – Anne Bauval Feb 07 '23 at 10:48
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    @AnneBauval Sure, usually I would talk about this with model theorists and would have just said "definable in the reals". I edited the question to be a bit more precise. – Florian Felix Feb 07 '23 at 15:30
  • Over coffee I discussed this with some colleagues and figure it might be possible to bound the computational complexity of periods and hence build some weird stuff like 0. Ackermann(1) zeros 1 Ackermann(2) zeros ..... – Florian Felix Feb 07 '23 at 15:32

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So, I found an answer to my question. This paper by Masahiko Yoshinaga, shows that periods are so called elementary real numbers, which are computable numbers that can be approximated by a fraction of so elementary functions, which is the smallest class of functions $f: \mathbb{N}^n \rightarrow \mathbb{N}$ (where $n$ is not fixed) that are closed under multiplication, addition, subtraction, composition and bounded sums and products. They also give and construct an example of a computable number, that is not elementary, hence not a period.

Florian Felix
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