Exponents of Diophantine Approximation and Sturmian Continued Fractions
Annales de l'Institut Fourier, Volume 55 (2005) no. 3, pp. 773-804.

Let ξ be a real number and let n be a positive integer. We define four exponents of Diophantine approximation, which complement the exponents w n (ξ) and w n * (ξ) defined by Mahler and Koksma. We calculate their six values when n=2 and ξ is a real number whose continued fraction expansion coincides with some Sturmian sequence of positive integers, up to the initial terms. In particular, we obtain the exact exponent of approximation to such a continued fraction ξ by quadratic surds.

Soient ξ un nombre réel et n un entier strictement positif. Nous définissons quatre exposants d’approximation diophantienne, qui viennent compléter les exposants w n (ξ) et w n * (ξ) définis par Mahler et Koksma. Nous calculons leurs six valeurs lorsque n=2 et ξ est un nombre réel dont le développement en fraction continue est, aux premiers termes près, une suite sturmienne d’entiers positifs. En particulier, nous obtenons l’exposant exact d’approximation d’une telle fraction continue ξ par des nombres quadratiques

DOI: 10.5802/aif.2114
Classification: 11J13, 11J82
Keywords: Diophantine approximation, Sturmian sequence, simultaneous approximation, transcendence measure
Mot clés : approximation diophantienne, suite sturmienne, approximation simultanée, mesure de transcendance
Bugeaud, Yann 1; Laurent, Michel 

1 Université Louis Pasteur, U. F. R. de mathématiques, 7 rue René Descartes, 67084 STRASBOURG (France), Institut de Mathématiques de Luminy, U.P.R. 9016, case 907, 163 avenue de Luminy, 13288 MARSEILLE CEDEX 9 (France)
@article{AIF_2005__55_3_773_0,
     author = {Bugeaud, Yann and Laurent, Michel},
     title = {Exponents of {Diophantine} {Approximation} and {Sturmian} {Continued} {Fractions}},
     journal = {Annales de l'Institut Fourier},
     pages = {773--804},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {55},
     number = {3},
     year = {2005},
     doi = {10.5802/aif.2114},
     mrnumber = {2149403},
     zbl = {1155.11333},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2114/}
}
TY  - JOUR
AU  - Bugeaud, Yann
AU  - Laurent, Michel
TI  - Exponents of Diophantine Approximation and Sturmian Continued Fractions
JO  - Annales de l'Institut Fourier
PY  - 2005
SP  - 773
EP  - 804
VL  - 55
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2114/
DO  - 10.5802/aif.2114
LA  - en
ID  - AIF_2005__55_3_773_0
ER  - 
%0 Journal Article
%A Bugeaud, Yann
%A Laurent, Michel
%T Exponents of Diophantine Approximation and Sturmian Continued Fractions
%J Annales de l'Institut Fourier
%D 2005
%P 773-804
%V 55
%N 3
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2114/
%R 10.5802/aif.2114
%G en
%F AIF_2005__55_3_773_0
Bugeaud, Yann; Laurent, Michel. Exponents of Diophantine Approximation and Sturmian Continued Fractions. Annales de l'Institut Fourier, Volume 55 (2005) no. 3, pp. 773-804. doi : 10.5802/aif.2114. http://www.numdam.org/articles/10.5802/aif.2114/

[1] W.W. Adams; J.L. Davison A remarkable class of continued fractions, Proc. Amer. Math. Soc., Volume 65 (1977), pp. 194-198 | DOI | MR | Zbl

[2] J.-P. Allouche; J.L. Davison; M. Quefféle ; L.Q. Zamboni Transcendence of Sturmian or morphic continued fractions, J. Number Theory, Volume 91 (2001), pp. 39-66 | DOI | MR | Zbl

[3] B. Arbour; D. Roy A Gel'fond type criterion in degree two, Acta Arith., Volume 11 (2004), pp. 97-103 | MR | Zbl

[4] A. Baker; W.M. Schmidt Diophantine approximation and Hausdorff dimension, Proc. London Math. Soc., Volume 21 (1970), pp. 1-11 | DOI | MR | Zbl

[5] R.C. Baker On approximation with algebraic numbers of bounded degree, Mathematika, Volume 23 (1976), pp. 18-31 | DOI | MR | Zbl

[6] V.I. Bernik Application of the Hausdorff dimension in the theory of Diophantine approximations, Acta Arith., Volume 42 (1983), pp. 219-253 | MR | Zbl

[7] Y. Bugeaud On the approximation by algebraic numbers with bounded degree, Algebraic number theory and Diophantine analysis (Graz, 1998) (2000), pp. 47-53 | Zbl

[8] Y. Bugeaud Approximation par des nombres algébriques, J. Number Theory, Volume 84 (2000), pp. 15-33 | DOI | MR | Zbl

[9] Y. Bugeaud Mahler's classification of numbers compared with Koksma's, Acta Arith., Volume 110 (2003), pp. 89-105 | DOI | MR | Zbl

[10] Y. Bugeaud Approximation by algebraic numbers, Cambridge Tracts in Math., 160, Cambridge University Press, 2004 | MR | Zbl

[11] Y. Bugeaud; O. Teulié Approximation d'un nombre réel par des nombres algébriques de degré donné, Acta Arith., Volume 93 (2000), pp. 77-86 | MR | Zbl

[12] J. Cassaigne Limit values of the recurrence quotient of Sturmian sequences, Theor. Comput. Sci., Volume 218 (1999), pp. 3-12 | DOI | MR | Zbl

[13] H. Davenport; W.M. Schmidt Approximation to real numbers by quadratic irrationals, Acta Arith., Volume 13 (1967), pp. 169-176 | MR | Zbl

[14] H. Davenport; W.M. Schmidt Approximation to real numbers by algebraic integers, Acta Arith., Volume 15 (1969), pp. 393-416 | MR | Zbl

[15] H. Davenport; W.M. Schmidt Dirichlet's theorem on Diophantine approximation (Symposia Mathematica (INDAM, Rome, 1968/69)), Volume IV (1970), pp. 113-132 | Zbl

[16] J.L. Davison A series and its associated continued fraction, Proc. Amer. Math. Soc., Volume 63 (1977), pp. 29-32 | DOI | MR | Zbl

[17] K. Falconer The geometry of fractal sets, Cambridge Tracts in Mathematics, 85, Cambridge University Press, 1985 | MR | Zbl

[18] V. JarniK Zur metrischen Theorie der diophantischen Approximationen, Prace Mat.-Fiz., Volume 36 (1928/29), pp. 91-106 | JFM

[19] V. JarniK Zum Khintchineschen `Übertragungssatz', Trav. Inst. Math. Tbilissi, Volume 3 (1938), pp. 193-212 | Zbl

[20] J.F. Koksma Über die Mahlersche Klasseneinteilung der transzendenten Zahlen und die Approximation komplexer Zahlen durch algebraische Zahlen, Monats. Math. Phys., Volume 48 (1939), pp. 176-189 | DOI | JFM | MR | Zbl

[21] M. Laurent Some remarks on the approximation of complex numbers by algebraic numbers, Proceedings of the 2nd Panhellenic Conference in Algebra and Number Theory (Thessaloniki, 1998) (Bull. Greek Math. Soc.), Volume 42 (1999), pp. 49-57 | Zbl

[22] M. Laurent Simultaneous rational approximation to the successive powers of a real number, Indag. Math., Volume 11 (2003), pp. 45-53 | MR | Zbl

[23] K. Mahler Zur Approximation der Exponentialfunktionen und des Logarithmus. I, II, J. reine angew. Math., Volume 166 (1932), pp. 118-150 | Zbl

[24] M. Queffélec Approximations diophantiennes des nombres sturmiens, J. Théor. Nombres Bordeaux, Volume 14 (2002), pp. 613-628 | DOI | Numdam | MR | Zbl

[25] A.M. Rockett; P. Szüsz Continued Fractions, World Scientific, Singapore, 1992 | MR | Zbl

[26] D. Roy Approximation simultanée d'un nombre et son carré, C. R. Acad. Sci. Paris, Volume 336 (2003), pp. 1-6 | MR | Zbl

[27] D. Roy Approximation to real numbers by cubic algebraic numbers, I, Proc. London Math. Soc., Volume 88 (2004), pp. 42-62 | DOI | MR | Zbl

[28] D. Roy Approximation to real numbers by cubic algebraic numbers, II, Ann. of Math., Volume 158 (2003), pp. 1081-1087 | DOI | MR | Zbl

[29] D. Roy Diophantine approximation in small degree (CRM Proceedings and Lecture Notes), Volume 36 (2004), pp. 269-285 | Zbl

[30] V.G. Sprindzuk Mahler's problem in metric number theory, 25, Amer. Math. Soc., Providence, R.I., 1969 | MR | Zbl

[31] E. Wirsing Approximation mit algebraischen Zahlen beschränkten Grades, J. reine angew. Math., Volume 206 (1961), pp. 67-77 | MR | Zbl

Cited by Sources: