Extensions of the Cugiani-Mahler theorem

Abstract

In 1955, Roth established that if ξ is an irrational number such that there are a positive real number e and infinitely many rational numbers p/q with q ≥ 1 and |ξ − p/q| < q−2−e , then ξ is transcendental. A few years later, Cugiani obtained the same conclusion with e replaced by a function q → e(q) that decreases very slowly to zero, provided that the sequence of rational solutions to |ξ − p/q| < q−2−e(q) 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. Mathematics Subject Classification (2000): 11J68.