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Comptes Rendus Mathématique
Volume 339, n° 10
pages 679-682 (novembre 2004)
Doi : 10.1016/j.crma.2004.10.009
Received : 13 February 2004 ;  accepted : 4 October 2004
Spectres pour l'approximation d'un nombre réel et de son carré
Spectra for approximating a real number and its square.

Stéphane Fischler
D.M.A., E.N.S., 45, rue d'Ulm, 75230 Paris cedex 05, France 


Notons   l'exposant qui mesure comment un nombre réel non quadratique et son carré peuvent être approchés simultanément par des nombres rationnels de même dénominateur. Davenport et Schmidt ont démontré que   est toujours compris (au sens large) entre le nombre d'or γ et 2. Roy, puis Bugeaud et Laurent, ont construit à l'aide de mots ayant beaucoup de préfixes palindromes des réels tels que  . Dans ce texte, on définit de nouveaux exposants d'approximation qui permettent, dans une certaine mesure, de caractériser les valeurs de   obtenues par ces auteurs. Pour citer cet article : S. Fischler, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

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Let   be the exponent that measures how a non-quadratic real number and its square can be simultaneously approximated by rational numbers with the same denominator. Davenport and Schmidt have proved that   is always between the golden ratio γ and 2. Roy, and after him Bugeaud and Laurent, have constructed numbers such that  . Their method involves infinite words with many palindrome prefixes. In this text, we define new exponents of approximation that allow us to obtain, to some extent, a characterization of the values   obtained by these authors. To cite this article: S. Fischler, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

The full text of this article is available in PDF format.

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