Adamczewski, Boris; Bugeaud, Yann; Davison, Les Continued fractions and transcendental numbers. (English) Zbl 1152.11034 Ann. Inst. Fourier 56, No. 7, 2093-2113 (2006). The paper deals with the several criteria for continued fractions to be transcendental. Some applications in the direction to Davison’s, Rudin-Shapiro or automatic continued fractions are included. The proofs make use some elements from the combinatorial theory. Reviewer: Jaroslav Hančl (Ostrava) Cited in 6 Documents MSC: 11J81 Transcendence (general theory) 11J70 Continued fractions and generalizations 68R15 Combinatorics on words Keywords:transcendence; continued fraction Citations:Zbl 1195.11093 PDFBibTeX XMLCite \textit{B. Adamczewski} et al., Ann. Inst. Fourier 56, No. 7, 2093--2113 (2006; Zbl 1152.11034) Full Text: DOI arXiv Numdam EuDML References: [1] Adamczewski, B., Transcendance “à la Liouville” de certain nombres réels, C. R. Acad. Sci. Paris, 338, 511-514 (2004) · Zbl 1046.11051 [2] Adamczewski, B.; Bugeaud, Y., On the complexity of algebraic numbers I. Expansions in integer bases · Zbl 1195.11094 [3] Adamczewski, B.; Bugeaud, Y., On the complexity of algebraic numbers II. Continued fractions · Zbl 1195.11093 [4] Adamczewski, B.; Bugeaud, Y., On the Maillet-Baker continued fractions · Zbl 1145.11054 [5] Adamczewski, B.; Bugeaud, Y.; Luca, F., Sur la complexité des nombres algébriques, C. R. Acad. Sci. Paris, 339, 11-14 (2004) · Zbl 1119.11019 [6] Allouche, J.-P., Nouveaux résultats de transcendance de réels à développements non aléatoire, Gaz. Math., 84, 19-34 (2000) [7] Allouche, J.-P.; Davison, J. L.; Queffélec, M.; Zamboni, L. Q., Transcendence of Sturmian or morphic continued fractions, J. Number Theory, 91, 39-66 (2001) · Zbl 0998.11036 [8] Allouche, J.-P.; Shallit, J. O., Generalized Pertured Symmetry, European J. Combin., 19, 401-411 (1998) · Zbl 0918.11015 [9] Allouche, J.-P.; Shallit, J. O., Automatic Sequences: Theory, Applications, Generalizations (2003) · Zbl 1086.11015 [10] Baker, A., Continued fractions of transcendental numbers, Mathematika, 9, 1-8 (1962) · Zbl 0105.03903 [11] Baker, A., On Mahler’s classification of transcendental numbers, Acta Math., 111, 97-120 (1964) · Zbl 0147.03403 [12] Baum, L. E.; Sweet, M. M., Continued fractions of algebraic power series in characteristic \(2\), Ann. of Math., 103, 593-610 (1976) · Zbl 0312.10024 [13] Baxa, C., Extremal values of continuants and transcendence of certain continued fractions, Adv. in Appl. Math., 32, 754-790 (2004) · Zbl 1063.11019 [14] Blanchard, A.; Mendès France, M., Symétrie et transcendance, Bull. Sci. Math., 106, 325-335 (1982) · Zbl 0492.10027 [15] Davenport, H.; Roth, K. F., Rational approximations to algebraic numbers, Mathematika, 2, 160-167 (1955) · Zbl 0066.29302 [16] Davison, J. L., A class of transcendental numbers with bounded partial quotients, Theory and Applications, R. A. Mollin, ed., 365-371 (1989) · Zbl 0693.10028 [17] Davison, J. L., Continued fractions with bounded partial quotients, Proc. Edinb. Math. Soc., 45, 653-671 (2002) · Zbl 1107.11303 [18] Dekking, F. M.; Mendès France, M.; van der Poorten, A. J., Folds!, Math. Intelligencer, 4, 130-138, 173-181, 190-195 (1982) · Zbl 0493.10003 [19] Evertse, J.-H., The number of algebraic numbers of given degree approximating a given algebraic number, London Math. Soc. Lecture Note Ser. 247, In Analytic number theory (Kyoto, 1996), 53-83 (1997) · Zbl 0919.11048 [20] Fogg, N. Pytheas, Substitutions in Dynamics, Arithmetics and Combinatorics (2002) · Zbl 1014.11015 [21] Hartmanis, J.; Stearns, R. E., On the computational complexity of algorithms, Trans. Amer. Math. Soc., 117, 285-306 (1965) · Zbl 0131.15404 [22] Khintchine, A. Ya., Continued fractions, 2nd edition (1949) · Zbl 0117.28503 [23] Lang, S., Introduction to Diophantine Approximations (1995) · Zbl 0826.11030 [24] LeVeque, W. J., Topics in number theory, Vol. II (1956) · Zbl 0070.03803 [25] Liardet, P.; Stambul, P., Séries de Engel et fractions continuées, J. Théor. Nombres Bordeaux, 12, 37-68 (2000) · Zbl 1007.11045 [26] 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, 18, 883-885, 910-911 (1844) [27] Loxton, J. H.; van der Poorten, A. J., Arithmetic properties of certain functions in several variables III, Bull. Austral. Math. Soc., 16, 15-47 (1977) · Zbl 0339.10028 [28] Maillet, E., Introduction à la théorie des nombres transcendants et des propriétés arithmétiques des fonctions (1906) · JFM 37.0237.02 [29] Mendès France, M., Principe de la symétrie perturbée, Séminaire de Théorie des Nombres, Paris 1979-80, M.-J. Bertin (éd.), Birkhäuser, Boston, 77-98 (1981) · Zbl 0451.10019 [30] Perron, O., Die Lehre von den Kettenbrüchen (1929) · JFM 55.0262.09 [31] van der Poorten, A. J.; Shallit, J. O., Folded continued fractions, J. Number Theory, 40, 237-250 (1992) · Zbl 0753.11005 [32] Queffélec, M., Transcendance des fractions continues de Thue-Morse, J. Number Theory, 73, 201-211 (1998) · Zbl 0920.11045 [33] Queffélec, M., Irrational number with automaton-generated continued fraction expansion, World Scientific, Dynamical Systems: From Crystal to Chaos, 190-198 (2000) · Zbl 1196.11015 [34] Ridout, D., Rational approximations to algebraic numbers, Mathematika, 4, 125-131 (1957) · Zbl 0079.27401 [35] Rudin, W., Some theorems on Fourier coefficients, Proc. Amer. Math. Soc., 10, 855-859 (1959) · Zbl 0091.05706 [36] Schmidt, W. M., On simultaneous approximations of two algebraic numbers by rationals, Acta Math., 119, 27-50 (1967) · Zbl 0173.04801 [37] Schmidt, W. M., Norm form equations, Ann. of Math., 96, 526-551 (1972) · Zbl 0226.10024 [38] Schmidt, W. M., Diophantine approximation (1980) · Zbl 0421.10019 [39] Shallit, J. O., Real numbers with bounded partial quotients: a survey, Enseign. Math., 38, 151-187 (1992) · Zbl 0753.11006 [40] Shapiro, H. S., Extremal problems for polynomials and power series (1952) [41] Waldschmidt, M., Un demi-siècle de transcendance, Development of mathematics 1950-2000, 1121-1186 (2000) · Zbl 0977.11030 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.