Let $\alpha \in \mathbb{ C}$ be an exponential period. 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.
Categories: math.NT