Extensions of the Cugiani-Mahler theorem
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 6 (2007) no. 3, p. 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.
@article{ASNSP_2007_5_6_3_477_0,
     author = {Bugeaud, Yann},
     title = {Extensions of the Cugiani-Mahler theorem},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 6},
     number = {3},
     year = {2007},
     pages = {477-498},
     zbl = {1139.11032},
     mrnumber = {2370270},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2007_5_6_3_477_0}
}
Bugeaud, Yann. Extensions of the Cugiani-Mahler theorem. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 6 (2007) no. 3, pp. 477-498. http://www.numdam.org/item/ASNSP_2007_5_6_3_477_0/

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