A determinantal approach to irrationality

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 my talk I
give an extension of this criterion in the case when the sequence possesses an
additional `period' structure; in particular, the requirement $q_n\xi-p_n\to 0$ is
weakened. Some applications are discussed including a new proof of the
irrationality of $\pi$.