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
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.2114/

[1] W.W. Adams; J.L. Davison A remarkable class of continued fractions, Proc. Amer. Math. Soc., Tome 65 (1977), pp. 194-198 | Article | MR: 441879 | Zbl: 0366.10027

[2] J.-P. Allouche; J.L. Davison; M. Quefféle ; L.Q. Zamboni Transcendence of Sturmian or morphic continued fractions, J. Number Theory, Tome 91 (2001), pp. 39-66 | Article | MR: 1869317 | Zbl: 0998.11036

[3] B. Arbour; D. Roy A Gel'fond type criterion in degree two, Acta Arith., Tome 11 (2004), pp. 97-103 | MR: 2038064 | Zbl: 1064.11049

[4] A. Baker; W.M. Schmidt Diophantine approximation and Hausdorff dimension, Proc. London Math. Soc., Tome 21 (1970), pp. 1-11 | Article | MR: 271033 | Zbl: 0206.05801

[5] R.C. Baker On approximation with algebraic numbers of bounded degree, Mathematika, Tome 23 (1976), pp. 18-31 | Article | MR: 409373 | Zbl: 0327.10034

[6] V.I. Bernik Application of the Hausdorff dimension in the theory of Diophantine approximations, Acta Arith., Tome 42 (1983), pp. 219-253 | MR: 729734 | Zbl: 0482.10049

[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: 0959.11033

[8] Y. Bugeaud Approximation par des nombres algébriques, J. Number Theory, Tome 84 (2000), pp. 15-33 | Article | MR: 1782258 | Zbl: 0967.11025

[9] Y. Bugeaud Mahler's classification of numbers compared with Koksma's, Acta Arith., Tome 110 (2003), pp. 89-105 | Article | MR: 2007546 | Zbl: 1029.11034

[10] Y. Bugeaud Approximation by algebraic numbers, Cambridge Tracts in Math., Tome 160, Cambridge University Press, 2004 | MR: 2136100 | Zbl: 1055.11002

[11] Y. Bugeaud; O. Teulié Approximation d'un nombre réel par des nombres algébriques de degré donné, Acta Arith., Tome 93 (2000), pp. 77-86 | MR: 1760090 | Zbl: 0948.11029

[12] J. Cassaigne Limit values of the recurrence quotient of Sturmian sequences, Theor. Comput. Sci., Tome 218 (1999), pp. 3-12 | Article | MR: 1687748 | Zbl: 0916.68115

[13] H. Davenport; W.M. Schmidt Approximation to real numbers by quadratic irrationals, Acta Arith., Tome 13 (1967), pp. 169-176 | MR: 219476 | Zbl: 0155.09503

[14] H. Davenport; W.M. Schmidt Approximation to real numbers by algebraic integers, Acta Arith., Tome 15 (1969), pp. 393-416 | MR: 246822 | Zbl: 0186.08603

[15] H. Davenport; W.M. Schmidt Dirichlet's theorem on Diophantine approximation (Symposia Mathematica (INDAM, Rome, 1968/69)) Tome IV (1970), pp. 113-132 | Zbl: 0226.10032

[16] J.L. Davison A series and its associated continued fraction, Proc. Amer. Math. Soc., Tome 63 (1977), pp. 29-32 | Article | MR: 429778 | Zbl: 0326.10030

[17] K. Falconer The geometry of fractal sets, Cambridge Tracts in Mathematics, Tome 85, Cambridge University Press, 1985 | MR: 867284 | Zbl: 0587.28004

[18] V. JarniK Zur metrischen Theorie der diophantischen Approximationen, Prace Mat.-Fiz., Tome 36 (1928/29), pp. 91-106 | JFM: 55.0718.01

[19] V. JarniK Zum Khintchineschen `Übertragungssatz', Trav. Inst. Math. Tbilissi, Tome 3 (1938), pp. 193-212 | Zbl: 0019.10602

[20] J.F. Koksma Über die Mahlersche Klasseneinteilung der transzendenten Zahlen und die Approximation komplexer Zahlen durch algebraische Zahlen, Monats. Math. Phys., Tome 48 (1939), pp. 176-189 | Article | JFM: 65.0180.01 | MR: 845 | Zbl: 0021.20804

[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.) Tome 42 (1999), pp. 49-57 | Zbl: 0971.11035

[22] M. Laurent Simultaneous rational approximation to the successive powers of a real number, Indag. Math., Tome 11 (2003), pp. 45-53 | MR: 2015598 | Zbl: 1049.11069

[23] K. Mahler Zur Approximation der Exponentialfunktionen und des Logarithmus. I, II, J. reine angew. Math., Tome 166 (1932), pp. 118-150 | Zbl: 0003.15101

[24] M. Queffélec Approximations diophantiennes des nombres sturmiens, J. Théor. Nombres Bordeaux, Tome 14 (2002), pp. 613-628 | Article | Numdam | MR: 2040697 | Zbl: 02184603

[25] A.M. Rockett; P. Szüsz Continued Fractions, World Scientific, Singapore, 1992 | MR: 1188878 | Zbl: 0925.11038

[26] D. Roy Approximation simultanée d'un nombre et son carré, C. R. Acad. Sci. Paris, Tome 336 (2003), pp. 1-6 | MR: 1968892 | Zbl: 1038.11042

[27] D. Roy Approximation to real numbers by cubic algebraic numbers, I, Proc. London Math. Soc., Tome 88 (2004), pp. 42-62 | Article | MR: 2018957 | Zbl: 1035.11028

[28] D. Roy Approximation to real numbers by cubic algebraic numbers, II, Ann. of Math., Tome 158 (2003), pp. 1081-1087 | Article | MR: 2031862 | Zbl: 1044.11061

[29] D. Roy Diophantine approximation in small degree (CRM Proceedings and Lecture Notes) Tome 36 (2004), pp. 269-285 | Zbl: 02152663

[30] V.G. Sprindzuk Mahler's problem in metric number theory Tome 25, Amer. Math. Soc., Providence, R.I., 1969 | MR: 245527 | Zbl: 0181.05502

[31] E. Wirsing Approximation mit algebraischen Zahlen beschränkten Grades, J. reine angew. Math., Tome 206 (1961), pp. 67-77 | MR: 142510 | Zbl: 0097.03503

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