Extensions of the Cugiani-Mahler theorem

Bugeaud, Yann

Bugeaud, Yann

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 6 (2007) no. 3, p. 477-498

Abstract

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

q \geq 1

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

q \mapsto ε (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.

MR 2370270 | Zbl 1139.11032 | 1 citation in Numdam

Classification: 11J68

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     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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