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?
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.