Extensions of the Cugiani-Mahler theorem
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 3, pp. 477-498.

In 1955, Roth established that if ξ is an irrational number such that there are a positive real number ε and infinitely many rational numbers p/q with q1 and |ξ-p/q|<q -2-ε , then ξ is transcendental. A few years later, Cugiani obtained the same conclusion with ε replaced by a function qε(q) that decreases very slowly to zero, provided that the sequence of rational solutions to |ξ-p/q|<q -2-ε(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.

Classification : 11J68
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Bugeaud, Yann. Extensions of the Cugiani-Mahler theorem. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 3, pp. 477-498. http://www.numdam.org/item/ASNSP_2007_5_6_3_477_0/

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