A determinantal approach to irrationality

Abstract

It is a classical fact that the irrationality of a number $\xi\in\mathbb R$ follows from the existence of a sequence $p_n/q_n$ with integral $p_n$ and $q_n$ such that $q_n\xi-p_n\ne0$ for all $n$ and $q_n\xi-p_n\to0$ as $n\to\infty$. In this note we give an extension of this criterion in the case when the sequence possesses an additional structure; in particular, the requirement $q_n\xi-p_n\to0$ is weakened. Some applications are given including a new proof of the irrationality of $\pi$. Finally, we discuss analytical obstructions to extend the new irrationality criterion further and speculate about some mathematical constants whose irrationality is still to be established.