Abstract

In 1955, Roth established that if $\xi $ is an irrational number such that there are a positive real number $\varepsilon $ and infinitely many rational numbers $p/q$ with $q \ge 1$ and $|\xi - p/q| < q^{-2-\varepsilon }$, then $\xi $ is transcendental. A few years later, Cugiani obtained the same conclusion with $\varepsilon $ replaced by a function $q \mapsto \varepsilon $ that decreases very slowly to zero, provided that the sequence of rational solutions to $|\xi - p/q| < q^{-2-\varepsilon }$ is sufficiently dense, in a suitable sense. We give an alternative, and much simpler, proof of Cugiani’s Theorem and extend it to simultaneous approximation