Hankel determinants and irrationality questions

It is a classical fact that the irrationality of a real number $x$ follows from the existence of a sequence $p_n/q_n$, with integral $p_n$ and $q_n$, such that $q_nx-p_n$ is nonzero for all $n$ and tends to $0$ as $n$ tends to infinity. In my talk I will discuss an extension of this criterion in the case when the sequence possesses an additional structure; in particular, the requirement $q_nx-p_n\to 0$ is weakened. Some applications will be given including a new proof of the irrationality of $\pi$.