Approximations diophantiennes des nombres sturmiens
Journal de théorie des nombres de Bordeaux, Tome 14 (2002) no. 2, pp. 613-628.

Nous établissons pour tout nombre sturmien (de développement dyadique sturmien) des propriétés d'approximation diophantienne très précises, ne dépendant que de l'angle de la suite sturmienne, généralisant ainsi des travaux antérieurs de Ferenczi-Mauduit et Bullett-Sentenac.

Generalizing previous results of Ferenczi-Mauduit and Bullett-Sentenac, we prove that any sturmian number (with sturmian dyadic expansion) enjoys very sharp diophantine approximation properties, depending only on the angle of the sturmian sequence.

@article{JTNB_2002__14_2_613_0,
     author = {Martine Queff\'elec},
     title = {Approximations diophantiennes des nombres sturmiens},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {613--628},
     publisher = {Universit\'e Bordeaux I},
     volume = {14},
     number = {2},
     year = {2002},
     zbl = {1076.11044},
     mrnumber = {2040697},
     language = {fr},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_2002__14_2_613_0/}
}
TY  - JOUR
AU  - Martine Queffélec
TI  - Approximations diophantiennes des nombres sturmiens
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2002
SP  - 613
EP  - 628
VL  - 14
IS  - 2
PB  - Université Bordeaux I
UR  - https://jtnb.centre-mersenne.org/item/JTNB_2002__14_2_613_0/
LA  - fr
ID  - JTNB_2002__14_2_613_0
ER  - 
%0 Journal Article
%A Martine Queffélec
%T Approximations diophantiennes des nombres sturmiens
%J Journal de théorie des nombres de Bordeaux
%D 2002
%P 613-628
%V 14
%N 2
%I Université Bordeaux I
%U https://jtnb.centre-mersenne.org/item/JTNB_2002__14_2_613_0/
%G fr
%F JTNB_2002__14_2_613_0
Martine Queffélec. Approximations diophantiennes des nombres sturmiens. Journal de théorie des nombres de Bordeaux, Tome 14 (2002) no. 2, pp. 613-628. https://jtnb.centre-mersenne.org/item/JTNB_2002__14_2_613_0/

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

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

[3] J.-P. Allouche, L.Q. Zamboni, Algebraic irrational binary numbers cannot be fixed points of non-trivial constant length or primitive morphisms. J. Number Theory 69 (1998), 119-124. | MR | Zbl

[4] P. Arnoux, G. Rauzy, Représentation géométrique de suites de complexité 2n + 1. Bull. Soc. Math. France 119 (1991), 199-215. | Numdam | MR | Zbl

[5] V. Berthé, Fréquences des facteurs des suites sturmiennes. Theoret. Comput. Sci. 165 (1996), 295-309. | MR | Zbl

[6] P.E. Böhmer, Über die Transzendenz gewisser dyadischer Brüche. Math. Ann. 96 (1926), 367-377. Erratum ibid. page 735. | JFM | MR

[7] T. Bousch, Le poisson n'a pas d'arête. Ann. Inst. H. Poincaré Probab. Stat. 36 (2000) 489-508. | Numdam | MR | Zbl

[8] S. Bullett, P. Sentenac, Ordered orbits of the shift, square roots, and the devil's staircase. Math. Proc. Camb. Philos. Soc. 115 (1994), 451-481. | MR | Zbl

[9] L.V. Danilov, Some classes of transcendent al numbers. English Translation in Math. Notes Acad. Sci. USSR 12 (1972), 524-527. | MR | Zbl

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

[11] F.M. Dekking, Transcendance du nombre de Thue-Morse. C. R. Acad. Sci. Paris, Sér. A-B 285 (1977), 157-160. | MR | Zbl

[12] F.M. Dekking, On the Thue-Morse measure. Acta Universitatis Carolinae, Math. Phys. 33 (1992), 35-40. | MR | Zbl

[13] S. Ferenczi, C. Mauduit, Transcendence of numbers with a low complexity expansion. J. Number Theory 67 (1997), 146-161. | MR | Zbl

[14] G.H. Hardy, E.M. Wright, An introduction to the theory of numbers. Clarendon Press, Oxford Univ. Press, 1979. | MR | Zbl

[15] A. Hurwitz, Über die Kettenbruch-Entwicklung der Zahl e. Mathematische Werke, Bd 2 Basel, Birkhaüser, 1933, 129-133.

[16] T. Komatsu, A certain power series and the inhomogeneous continued fraction expansions. J. Number Theory 59 (1996), 291-312. | MR | Zbl

[17] P. Liardet, P. Stambul, Séries de Engel et fractions continuées. J. Théor. Nombres Bordeaux 12 (2000), 37-68. | Numdam | MR | Zbl

[18] J.H. Loxton, A.J. Van Der Poorten, Arithmetic properties of the solutions of a class of functional equations. J. Reine Angew. Math. 330 (1982), 159-172. | MR | Zbl

[19] M. Morse, G.A. Hedlund, Symbolic dynamics II: Sturmian trajectories. Amer. J. Math. 62 (1940), 1-42. | JFM | MR | Zbl

[20] K. Nishioka, I. Shiokawa, J.-I. Tamura, Arithmetical properties of a certain power series. J. Number Theory 42 (1992), 61-87. | MR | Zbl

[21] G.N. Raney, On continued fractions and finite automata. Math. Ann. 206 (1973), 265-283. | MR | Zbl

[22] D. Ridout, Rational approximations to algebraic numbers. Mathematika 4 (1957), 125-131. | MR | Zbl

[23] K.F. Roth, Rational approximations to algebraic numbers. Mathematika 2 (1955), 1-20. Corrigendum, page 168. | MR | Zbl

[24] R.N. Risley, L.Q. Zamboni, A generalization of Sturmian sequences: combinatorial structure and transcendence. Acta Arith. 95 (2000) 167-184. | MR | Zbl

[25] J. Shallit, Real numbers with bounded partial quotients: a survey. Enseign. Math. 38 (1992), 151-187. | MR | Zbl

[26] H.J.S. Smith, Note on continued fractions. Messenger Math. 6 (1876), 1-14. | JFM

[27] J.-I. Tamura, A class of transcendental numbers having explicit g-adic and Jacobi-Perron expansions of arbitrary dimension. Acta Arith. 71 (1995), 301-329. | MR | Zbl

[28] A.J. Van Der Poorten, An introduction to continued fractions. In Diophantine Analysis London Math. Soc. Lecture Note Ser., 109, Cambridge University Press, 1986, 99-138. | MR | Zbl