A determinantal approach to irrationality

It is a classical fact that the irrationality of a number ξ∈R follows from the existence of a sequence p_n/q_n with integral p_n and q_n such that q_nξ−p_n≠0 for all n and q_nξ−p_n→0 as n→∞. In this paper, we give an extension of this criterion in the case when the sequence possesses an additional structure; in particular, the requirement q_nξ−p_n→0 is weakened. Some applications are given, including a new proof of the irrationality of π. 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.