Palindromic continued fractions
[Fractions continues palindromes]
Adamczewski, Boris ; Bugeaud, Yann
Annales de l'Institut Fourier, Tome 57 (2007), p. 1557-1574 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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.

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.

Publié le : 2007-01-01
DOI : https://doi.org/10.5802/aif.2306
Classification:  11J81,  11J70,  68R15
Mots 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},
     volume = {57},
     year = {2007},
     pages = {1557-1574},
     doi = {10.5802/aif.2306},
     zbl = {1126.11036},
     mrnumber = {2364142},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2007__57_5_1557_0}
}

Adamczewski, Boris; Bugeaud, Yann. Palindromic continued fractions. Annales de l'Institut Fourier, Tome 57 (2007) pp. 1557-1574. doi : 10.5802/aif.2306. http://gdmltest.u-ga.fr/item/AIF_2007__57_5_1557_0/

[1] Adamczewski, B.; Bugeaud, Y. A Short Proof of the Transcendence of Thue-Morse Continued Fractions, Amer. Math. Monthly (to appear) | MR 2321257 | Zbl 1132.11330

[2] Adamczewski, B.; Bugeaud, Y. On the Maillet-Baker continued fractions, J. Reine Angew. Math. (to appear) | Zbl 1145.11054

[3] Adamczewski, B.; Bugeaud, Y. On the complexity of algebraic numbers, II. Continued fractions, Acta Math., Tome 195 (2005), pp. 1-20 | Article | MR 2233683 | Zbl 05039002

[4] Adamczewski, B.; Bugeaud, Y. On the Littlewood conjecture in simultaneous Diophantine approximation, J. London Math. Soc., Tome 73 (2006), pp. 355-366 | Article | MR 2225491 | Zbl 1093.11052

[5] Adamczewski, B.; Bugeaud, Y. Real and p-adic expansions involving symmetric patterns, Int. Math. Res. Not. (2006), pp. 17 | Article | MR 2250005 | Zbl 05122739

[6] Adamczewski, B.; Bugeaud, Y.; Davison, L. Continued fractions and transcendental numbers, Ann. Inst. Fourier (to appear) | Numdam

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

[8] Baker, A. Continued fractions of transcendental numbers, Mathematika, Tome 9 (1962), pp. 1-8 | Article | MR 144853 | Zbl 0105.03903

[9] Baker, A. On Mahler’s classification of transcendental numbers, Acta Math., Tome 111 (1964), pp. 97-120 | Article | Zbl 0147.03403

[10] Bugeaud, Y.; Laurent, M. Exponents of Diophantine and Sturmian continued fractions, Ann. Inst. Fourier, Tome 55 (2005), pp. 773-804 | Article | Numdam | MR 2149403 | Zbl 02171525

[11] Davison, J. L.; R. A. Mollin A class of transcendental numbers with bounded partial quotients, Number Theory and Applications, Kluwer Academic Publishers (1989), pp. 365-371 | MR 1123082 | Zbl 0693.10028

[12] Fischler, S. Palindromic Prefixes and Diophantine Approximation, Monatsh. Math. (to appear) | MR 2317388 | Zbl 1124.11032

[13] Fischler, S. Palindromic Prefixes and Episturmian Words, J. Combin. Theory, Ser. A, Tome 113 (2006), pp. 1281-1304 | Article | MR 2259061 | Zbl 1109.68082

[14] Jarník, V. Über die simultanen Diophantische Approximationen, Math. Z., Tome 33 (1931), pp. 505-543 | Article | MR 1545226

[15] Khintchine, A. Ya. Continued Fractions, Gosudarstv. Izdat. Tehn.-Theor. Lit., Moscow-Leningrad (1949) ((Russian) ; English translation: The University of Chicago Press, Chicago-London, 1964) | MR 44586

[16] Lang, S. Introduction to Diophantine Approximations, Springer-Verlag, New-York (1995) | MR 1348400 | Zbl 0826.11030

[17] Liouville, J. 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, Tome 19 (1844), p. 883-885 and 119–995

[18] Maillet, E. Introduction à la théorie des nombres transcendants et des propriétés arithmétiques des fonctions, Gauthier-Villars, Paris (1906)

[19] Perron, O. Die Lehre von den Ketterbrüchen, Teubner, Leipzig (1929)

[20] Queffélec, M. Transcendance des fractions continues de Thue–Morse, J. Number Theory, Tome 73 (1998), pp. 201-211 | Article | MR 1658023 | Zbl 0920.11045

[21] Roy, D. Approximation to real numbers by cubic algebraic integers, II, Ann. of Math. (2), Tome 158 (2003), pp. 1081-1087 | Article | MR 2031862 | Zbl 1044.11061

[22] Roy, D. Approximation to real numbers by cubic algebraic integers, I, Proc. London Math. Soc. (3), Tome 88 (2004), pp. 42-62 | Article | MR 2018957 | Zbl 1035.11028

[23] Schmidt, W. M. On simultaneous approximations of two algebraic numbers by rationals, Acta Math., Tome 119 (1967), pp. 27-50 | Article | MR 223309 | Zbl 0173.04801

[24] Schmidt, W. M. Norm form equations, Ann. of Math., Tome 96 (1972), pp. 526-551 | Article | MR 314761 | Zbl 0226.10024

[25] Schmidt, W. M. Diophantine approximation, Springer-Verlag, Berlin, Lecture Notes in Mathematics 785 (1980) | Zbl 0421.10019