Hankel determinants and irrationality questions

Abstract: 

It is a classical fact that the irrationality of a real number xx follows from the existence of a sequence pn/qnpn/qn, with integral pnpn and qnqn, such that qnx−pnqnx−pn is nonzero for all nn and tends to 00 as nn 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 qnx−pn→0qnx−pn→0 is weakened. Some applications will be given including a new proof of the irrationality of ππ.