On the Maillet-Baker continued fractions

Abstract We use the Schmidt Subspace Theorem to establish the transcendence of a class of quasi-periodic continued fractions. This improves earlier works of Maillet and of A. Baker. We also improve an old result of Davenport and Roth on the rate of increase of the denominators of the convergents to any real algebraic number.

[1]  A. Y. Khinchin.  Continued Fractions (In Russian) , 1937 .

[2]  J. Liouville.  Sur des classes très-étendues de quantités dont la valeur n'est ni algébrique, ni même réductible à des irrationnelles algébriques. , 1851 .

[3]  Yann Bugeaud,et al.  Palindromic continued fractions , 2005, math/0512014.

[4]  Y. Bugeaud,et al.  On the complexity of algebraic numbers I. Expansions in integer bases , 2005, math/0511674.

[5]  M. Waldschmidt.  Un demi siècle de transcendance , 2000 .

[6]  A. Baker.  On Mahler's classification of transcendental numbers , 1964 .

[7]  J. Davison,et al.  Quasi-periodic continued fractions , 2007 .

[8]  K. F. Roth,et al.  Rational approximations to algebraic numbers , 1955 .

[9]  Enrico Bombieri,et al.  Some quantitative results related to Roth's Theorem , 1988 .

[10]  H. Tietze.  Introduction à la théorie des nombres transcendants et des propriétés arithmétiques des fonctions , 1908 .

[11]  Analytic Number Theory: The Number of Algebraic Numbers of Given Degree Approximating a Given Algebraic Number , 1997 .

[12]  Y. Bugeaud,et al.  On the complexity of algebraic numbers , II . Continued fractions by , 2006 .

[13]  Maurice Mignotte.  Quelques remarques sur l'approximation rationnelle des nombres algébriques. , 1974 .

[14]  D. Ridout.  Rational approximations to algebraic numbers , 1957 .

[15]  Yann Bugeaud,et al.  Approximation by Algebraic Numbers , 2004 .

[16]  W. Leveque.  Topics in number theory , 1956 .

[17]  Wolfgang M. Schmidt,et al.  Norm Form Equations , 1972 .

[18]  A. Baker,et al.  Continued fractions of transcendental numbers , 1962 .

[19]  S. Lang,et al.  Introduction to Diophantine Approximations , 1995 .

[20]  Alan Baker,et al.  DIOPHANTINE APPROXIMATION (Lecture Notes in Mathematics, 785) , 1981 .