  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.
