Extensions of the Cugiani-Mahler theorem

Yann Bugeaud

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2007)

  • Volume: 6, Issue: 3, page 477-498
  • ISSN: 0391-173X

Abstract

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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 ≥ 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 ↦ 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.

How to cite

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Bugeaud, Yann. "Extensions of the Cugiani-Mahler theorem." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 6.3 (2007): 477-498. <http://eudml.org/doc/272286>.

@article{Bugeaud2007,
abstract = {In 1955, Roth established that if $\xi $ is an irrational number such that there are a positive real number $\varepsilon $ and infinitely many rational numbers $p/q$ with $q \ge 1$ and $|\xi - p/q| &lt; q^\{-2-\varepsilon \}$, then $\xi $ is transcendental. A few years later, Cugiani obtained the same conclusion with $\varepsilon $ replaced by a function $q \mapsto \varepsilon (q)$ that decreases very slowly to zero, provided that the sequence of rational solutions to $|\xi - p/q| &lt; q^\{-2-\varepsilon (q)\}$ 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.},
author = {Bugeaud, Yann},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {3},
pages = {477-498},
publisher = {Scuola Normale Superiore, Pisa},
title = {Extensions of the Cugiani-Mahler theorem},
url = {http://eudml.org/doc/272286},
volume = {6},
year = {2007},
}

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
PB - Scuola Normale Superiore, Pisa
VL - 6
IS - 3
SP - 477
EP - 498
AB - In 1955, Roth established that if $\xi $ is an irrational number such that there are a positive real number $\varepsilon $ and infinitely many rational numbers $p/q$ with $q \ge 1$ and $|\xi - p/q| &lt; q^{-2-\varepsilon }$, then $\xi $ is transcendental. A few years later, Cugiani obtained the same conclusion with $\varepsilon $ replaced by a function $q \mapsto \varepsilon (q)$ that decreases very slowly to zero, provided that the sequence of rational solutions to $|\xi - p/q| &lt; q^{-2-\varepsilon (q)}$ 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.
LA - eng
UR - http://eudml.org/doc/272286
ER -

References

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