Exponential periods and o-minimality I

Johan Commelin, Philipp Habegger, Annette Huber

Published 2020-07-16Version 1

Let $\alpha \in \mathbb{C}$ be an exponential period. This is the first part of a pair of papers where we show that the real and imaginary part of $\alpha$ 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.