Le développement de Stern–Brocot d’un nombre réel est une suite finie ou infinie de symboles
The Stern–Brocot expansion of a real number is a finite or infinite sequence of symbols
Révisé le :
Accepté le :
Publié le :
Mots-clés : Stern–Brocot tree, continued fractions, quadratic forms, quadratic numbers, Sturmian sequences
@article{JTNB_2019__31_3_697_0, author = {Reutenauer, Christophe}, title = {On the {Stern{\textendash}Brocot} expansion of real numbers}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {697--722}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {31}, number = {3}, year = {2019}, doi = {10.5802/jtnb.1104}, language = {en}, url = {https://www.numdam.org/articles/10.5802/jtnb.1104/} }
TY - JOUR AU - Reutenauer, Christophe TI - On the Stern–Brocot expansion of real numbers JO - Journal de théorie des nombres de Bordeaux PY - 2019 SP - 697 EP - 722 VL - 31 IS - 3 PB - Société Arithmétique de Bordeaux UR - https://www.numdam.org/articles/10.5802/jtnb.1104/ DO - 10.5802/jtnb.1104 LA - en ID - JTNB_2019__31_3_697_0 ER -
%0 Journal Article %A Reutenauer, Christophe %T On the Stern–Brocot expansion of real numbers %J Journal de théorie des nombres de Bordeaux %D 2019 %P 697-722 %V 31 %N 3 %I Société Arithmétique de Bordeaux %U https://www.numdam.org/articles/10.5802/jtnb.1104/ %R 10.5802/jtnb.1104 %G en %F JTNB_2019__31_3_697_0
Reutenauer, Christophe. On the Stern–Brocot expansion of real numbers. Journal de théorie des nombres de Bordeaux, Tome 31 (2019) no. 3, pp. 697-722. doi : 10.5802/jtnb.1104. https://www.numdam.org/articles/10.5802/jtnb.1104/
[1] Markov’s Theorem and 100 years of the Uniqueness Conjecture, a mathematical journey from irrational numbers to perfect matchings, Springer, 2013 | Zbl
[2] Une caractérisation simple des nombres de Sturm, J. Théor. Nombres Bordeaux, Volume 10 (1998) no. 2, pp. 237-241 | DOI | Numdam | Zbl
[3] Automatic sequences. Theory, applications, generalizations, Cambridge University Press, 2003 | Zbl
[4] Morphismes de Sturm, Bull. Belg. Math. Soc. Simon Stevin, Volume 1 (1994) no. 2, pp. 175-189 | DOI | MR | Zbl
[5] Image par homographie de mots de Christoffel, Bull. Belg. Math. Soc. Simon Stevin, Volume 8 (2001) no. 2, pp. 241-255 | DOI | MR | Zbl
[6] Quelques mots sur la droite projective réelle, J. Théor. Nombres Bordeaux, Volume 5 (1993) no. 1, pp. 23-51 | DOI | Numdam | Zbl
[7] A characterization of the quadratic irrationals, Can. Math. Bull., Volume 34 (1991) no. 1, pp. 36-41 | DOI | MR | Zbl
[8] Binary quadratic forms. Classical theory and modern computations, Springer, 1989 | Zbl
[9] Lehrsätze über arithmetische Eigenschaften der Irrationalzahlen, Annali di Mat., Volume XV (1887), pp. 253-276 | DOI | Zbl
[10] Substitution invariant cutting sequences, J. Théor. Nombres Bordeaux, Volume 5 (1993) no. 1, pp. 123-137 | DOI | Numdam | MR | Zbl
[11] Introduction to the theory of numbers, Dover Publications, 1957 | Zbl
[12] Episturmian words and some contructions of de Luca and Rauzy, Theor. Comput. Sci., Volume 255 (2001) no. 1-2, pp. 539-553 | DOI | Zbl
[13] Sur l’approximation des incommensurables et les séries trigonométriques, C. R. Math. Acad. Sci. Paris, Volume 139 (1904), pp. 1019-1021 | Zbl
[14] Substitutions in Dynamics, Arithmetics and Combinatorics (Fogg, N. Pytheas; Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A., eds.), Lecture Notes in Mathematics, 1794, Springer, 2002 | MR | Zbl
[15] The classification of rational approximations, Proc. Lond. Math. Soc., Volume 17 (1918), pp. 247-258 | DOI | MR
[16] Concrete mathematics, Addison-Wesley Publishing Group, 1994 | Zbl
[17] Ueber die angenäherte Darstellung der Zahlen durch rationale Brüche, Math. Ann., Volume 44 (1894), pp. 417-436 | DOI | Zbl
[18] Ueber die Reduction der binären quadratischen Formen, Math. Ann., Volume 45 (1894), pp. 85-117 | DOI | Zbl
[19] On continued fractions, substitutions and characteristic sequences
[20] Episturmian words and episturmian morphisms, Theor. Comput. Sci., Volume 276 (2002) no. 1-2, pp. 281-313 | DOI | MR | Zbl
[21] Substitution invariant Beatty sequences, Jap. J. Math., Volume 22 (1996) no. 2, pp. 349-354 | DOI | MR | Zbl
[22] Algebraic combinatorics on words, Encyclopedia of Mathematics and Its Applications, 90, Cambridge University Press, 2002 | MR | Zbl
[23] Symbolic dynamics II: Sturmian Trajectories, Am. J. Math., Volume 62 (1940), pp. 1-42 | DOI | MR | Zbl
[24] Exact arithmetic on the Stern–Brocot tree, J. Discrete Algorithms, Volume 5 (2007) no. 2, pp. 356-379 | DOI | MR | Zbl
[25] Propriétés d’invariance des mots sturmiens, J. Théor. Nombres Bordeaux, Volume 9 (1997) no. 2, pp. 351-369 | DOI | MR | Zbl
[26] On continued fractions and finite automata, Math. Ann., Volume 206 (1973), pp. 265-283 | DOI | MR | Zbl
[27] Sturmian images of non Sturmian words and standard morphisms, Theor. Comput. Sci., Volume 711 (2018), pp. 92-104 | DOI | MR | Zbl
[28] Cours d’algèbre supérieure. I., éditions Jacques Gabay, 1992 | MR | Zbl
[29] Binary quadratic forms as dessins, J. Théor. Nombres Bordeaux, Volume 29 (2017) no. 2, pp. 445-469 | DOI | MR | Zbl
[30] Zetafunktionen und quadratische Körper, Eine Einführung in die Zahlentheorie, Springer, 1981 | Zbl
Cité par Sources :