A determinantal approach to irrationality

TL;DR

This paper extends classical irrationality criteria by incorporating additional structure in sequences approximating real numbers, providing new proofs and insights into the irrationality of constants like pi.

Contribution

It introduces a generalized irrationality criterion based on structured sequences, broadening the scope of classical approximation methods.

Findings

New proof of the irrationality of pi

Extended irrationality criterion with weaker convergence conditions

Discussion of limitations and open problems in irrationality proofs

Abstract

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.