In the present work, we investigate real numbers whose sequence of partial quotients enjoys some combinatorial properties involving the notion of palindrome. We provide three new transendence criteria, that apply to a broad class of continued fraction expansions, including expansions with unbounded partial quotients. Their proofs heavily depend on the Schmidt Subspace Theorem.
Dans cet article, nous considérons des nombres réels dont la suite des quotients partiels jouit de certaines propriétés de symétrie faisant intervenir la notion de palindrome. Nous obtenons trois nouveaux critères de transcendance s’appliquant à une grande classe de fractions continues, qu’elles soient à quotients partiels bornés ou non. Les démonstrations de ces résultats reposent sur le théorème du sous-espace de Schmidt.
Keywords: Continued fractions, palindromes, transcendental numbers, Subspace Theorem.
Mot clés : Fractions continues, palindromes, nombres transcendants, théorème du sous-espace.
@article{AIF_2007__57_5_1557_0, author = {Adamczewski, Boris and Bugeaud, Yann}, title = {Palindromic continued fractions}, journal = {Annales de l'Institut Fourier}, pages = {1557--1574}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {57}, number = {5}, year = {2007}, doi = {10.5802/aif.2306}, zbl = {1126.11036}, mrnumber = {2364142}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2306/} }
TY - JOUR AU - Adamczewski, Boris AU - Bugeaud, Yann TI - Palindromic continued fractions JO - Annales de l'Institut Fourier PY - 2007 SP - 1557 EP - 1574 VL - 57 IS - 5 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2306/ DO - 10.5802/aif.2306 LA - en ID - AIF_2007__57_5_1557_0 ER -
%0 Journal Article %A Adamczewski, Boris %A Bugeaud, Yann %T Palindromic continued fractions %J Annales de l'Institut Fourier %D 2007 %P 1557-1574 %V 57 %N 5 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2306/ %R 10.5802/aif.2306 %G en %F AIF_2007__57_5_1557_0
Adamczewski, Boris; Bugeaud, Yann. Palindromic continued fractions. Annales de l'Institut Fourier, Volume 57 (2007) no. 5, pp. 1557-1574. doi : 10.5802/aif.2306. http://archive.numdam.org/articles/10.5802/aif.2306/
[1] A Short Proof of the Transcendence of Thue-Morse Continued Fractions, Amer. Math. Monthly (to appear) | MR | Zbl
[2] On the Maillet-Baker continued fractions, J. Reine Angew. Math. (to appear) | Zbl
[3] On the complexity of algebraic numbers, II. Continued fractions, Acta Math., Volume 195 (2005), pp. 1-20 | DOI | MR | Zbl
[4] On the Littlewood conjecture in simultaneous Diophantine approximation, J. London Math. Soc., Volume 73 (2006), pp. 355-366 | DOI | MR | Zbl
[5] Real and -adic expansions involving symmetric patterns, Int. Math. Res. Not. (2006), pp. 17 | DOI | MR | Zbl
[6] Continued fractions and transcendental numbers, Ann. Inst. Fourier (to appear) | Numdam
[7] Transcendence of Sturmian or morphic continued fractions, J. Number Theory, Volume 91 (2001), pp. 39-66 | DOI | MR | Zbl
[8] Continued fractions of transcendental numbers, Mathematika, Volume 9 (1962), pp. 1-8 | DOI | MR | Zbl
[9] On Mahler’s classification of transcendental numbers, Acta Math., Volume 111 (1964), pp. 97-120 | DOI | Zbl
[10] Exponents of Diophantine and Sturmian continued fractions, Ann. Inst. Fourier, Volume 55 (2005), pp. 773-804 | DOI | Numdam | MR | Zbl
[11] A class of transcendental numbers with bounded partial quotients, Number Theory and Applications, Kluwer Academic Publishers (1989), pp. 365-371 | MR | Zbl
[12] Palindromic Prefixes and Diophantine Approximation, Monatsh. Math. (to appear) | MR | Zbl
[13] Palindromic Prefixes and Episturmian Words, J. Combin. Theory, Ser. A, Volume 113 (2006), pp. 1281-1304 | DOI | MR | Zbl
[14] Über die simultanen Diophantische Approximationen, Math. Z., Volume 33 (1931), pp. 505-543 | DOI | MR
[15] Continued Fractions, Gosudarstv. Izdat. Tehn.-Theor. Lit., Moscow-Leningrad, 1949 (Russian) ; English translation: The University of Chicago Press, Chicago-London, 1964 | MR
[16] Introduction to Diophantine Approximations, Springer-Verlag, New-York, 1995 | MR | Zbl
[17] Sur des classes très étendues de quantités dont la valeur n’est ni algébrique, ni même réductible à des irrationelles algébriques, C. R. Acad. Sci. Paris, Volume 19 (1844), p. 883-885 and 119–995
[18] Introduction à la théorie des nombres transcendants et des propriétés arithmétiques des fonctions, Gauthier-Villars, Paris, 1906
[19] Die Lehre von den Ketterbrüchen, Teubner, Leipzig, 1929
[20] Transcendance des fractions continues de Thue–Morse, J. Number Theory, Volume 73 (1998), pp. 201-211 | DOI | MR | Zbl
[21] Approximation to real numbers by cubic algebraic integers, II, Ann. of Math. (2), Volume 158 (2003), pp. 1081-1087 | DOI | MR | Zbl
[22] Approximation to real numbers by cubic algebraic integers, I, Proc. London Math. Soc. (3), Volume 88 (2004), pp. 42-62 | DOI | MR | Zbl
[23] On simultaneous approximations of two algebraic numbers by rationals, Acta Math., Volume 119 (1967), pp. 27-50 | DOI | MR | Zbl
[24] Norm form equations, Ann. of Math., Volume 96 (1972), pp. 526-551 | DOI | MR | Zbl
[25] Diophantine approximation, Lecture Notes in Mathematics 785, Springer-Verlag, Berlin, 1980 | Zbl
Cited by Sources: