In 1955, Roth established that if is an irrational number such that there are a positive real number and infinitely many rational numbers with and , then is transcendental. A few years later, Cugiani obtained the same conclusion with replaced by a function that decreases very slowly to zero, provided that the sequence of rational solutions to 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}, pages = {477--498}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 6}, number = {3}, year = {2007}, zbl = {1139.11032}, mrnumber = {2370270}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2007_5_6_3_477_0/} }
TY - JOUR AU - Bugeaud, Yann TI - Extensions of the Cugiani-Mahler theorem JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2007 DA - 2007/// SP - 477 EP - 498 VL - Ser. 5, 6 IS - 3 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2007_5_6_3_477_0/ UR - https://zbmath.org/?q=an%3A1139.11032 UR - https://www.ams.org/mathscinet-getitem?mr=2370270 LA - en ID - ASNSP_2007_5_6_3_477_0 ER -
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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