Exponential periods and o-minimality

Abstract

Let $\hat{I}\pm \in \mathbb{C}$ be an exponential period. We show that the real and imaginary part of $\hat{I}\pm$ are up to signs volumes of sets definable in the o-minimal structure generated by $\mathbb{Q}$, the real exponential function and $\sin {|}_{[0,1]}$. This is a weaker analogue of the precise characterisation of ordinary periods as numbers whose real and imaginary part are up to signs volumes of $\mathbb{Q}$-semi-algebraic sets. Furthermore, we define a notion of naive exponential periods and compare it to the existing notions using cohomological methods. This points to a relation between the theory of periods and o-minimal structures.

72 pages. To appear in Annales de l'Institut Fourier. The paper is a merger of "Exponential periods and o-minimality I" (v1 of this submission) and "Exponential periods and o-minimality II", previously arXiv:2007.08290