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Continued fractions with low complexity: transcendence measures and quadratic approximation

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  19 March 2012

Yann Bugeaud
Yann Bugeaud*
Affiliation:
Mathématiques, Université de Strasbourg, 7, rue René Descartes, 67084 Strasbourg, France (email: [email protected])
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Abstract

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We establish measures of non-quadraticity and transcendence measures for real numbers whose sequence of partial quotients has sublinear block complexity. The main new ingredient is an improvement of Liouville’s inequality giving a lower bound for the distance between two distinct quadratic real numbers. Furthermore, we discuss the gap between Mahler’s exponent w2 and Koksma’s exponent w*2.

Keywords

continued fractiontranscendenceSchmidt’s subspace theorem

MSC classification

Secondary: 11J70: Continued fractions and generalizations 11J82: Measures of irrationality and of transcendence 11J87: Schmidt Subspace Theorem and applications
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 3 , May 2012 , pp. 718 - 750
Copyright
Copyright © Foundation Compositio Mathematica 2012

References

[AB05]Adamczewski, B. and Bugeaud, Y., On the complexity of algebraic numbers, II. Continued fractions, Acta Math. 195 (2005), 1–20.CrossRefGoogle Scholar
[AB06]Adamczewski, B. and Bugeaud, Y., On the Littlewood conjecture in simultaneous Diophantine approximation, J. Lond. Math. Soc. (2) 73 (2006), 355–366.CrossRefGoogle Scholar
[AB07a]Adamczewski, B. and Bugeaud, Y., Dynamics for β-shifts and Diophantine approximation, Ergodic Theory Dynam. Systems 27 (2007), 1695–1711.CrossRefGoogle Scholar
[AB07b]Adamczewski, B. and Bugeaud, Y., On the complexity of algebraic numbers I. Expansions in integer bases, Ann. of Math. (2) 165 (2007), 547–565.CrossRefGoogle Scholar
[AB10a]Adamczewski, B. and Bugeaud, Y., Mesures de transcendance et aspects quantitatifs de la méthode de Thue–Siegel–Roth–Schmidt, Proc. Lond. Math. Soc. (3) 101 (2010), 1–31.CrossRefGoogle Scholar
[AB10b]Adamczewski, B. and Bugeaud, Y., Transcendence measures for continued fractions involving repetitive or symmetric patterns, J. Eur. Math. Soc. 12 (2010), 883–914.Google Scholar
[AB11]Adamczewski, B. and Bugeaud, Y., Nombres réels de complexité sous-linéaire: mesures d’irrationalité et de transcendance, J. Reine Angew. Math. 658 (2011), 65–98.Google Scholar
[ABL04]Adamczewski, B., Bugeaud, Y. and Luca, F., Sur la complexité des nombres algébriques, C. R. Acad. Sci. Paris 339 (2004), 11–14.CrossRefGoogle Scholar
[AC06]Adamczewski, B. and Cassaigne, J., Diophantine properties of real numbers generated by finite automata, Compositio Math. 142 (2006), 1351–1372.CrossRefGoogle Scholar
[AR09]Adamczewski, B. and Rivoal, T., Irrationality measures for some automatic real numbers, Math. Proc. Cambridge Philos. Soc. 147 (2009), 659–678.CrossRefGoogle Scholar
[ADQZ01]Allouche, J.-P., Davison, J. L., Queffélec, M. and Zamboni, L. Q., Transcendence of Sturmian or morphic continued fractions, J. Number Theory 91 (2001), 39–66.CrossRefGoogle Scholar
[AS03]Allouche, J.-P. and Shallit, J., Automatic sequences: theory, applications, generalizations (Cambridge University Press, Cambridge, 2003).CrossRefGoogle Scholar
[Bak64]Baker, A., On Mahler’s classification of transcendental numbers, Acta Math. 111 (1964), 97–120.CrossRefGoogle Scholar
[Bak76]Baker, R. C., On approximation with algebraic numbers of bounded degree, Mathematika 23 (1976), 18–31.CrossRefGoogle Scholar
[Bug03]Bugeaud, Y., Mahler’s classification of numbers compared with Koksma’s, Acta Arith. 110 (2003), 89–105.CrossRefGoogle Scholar
[Bug04a]Bugeaud, Y., Approximation by algebraic numbers, Cambridge Tracts in Mathematics, vol. 160 (Cambridge University Press, Cambridge, 2004).CrossRefGoogle Scholar
[Bug04]Bugeaud, Y., Mahler’s classification of numbers compared with Koksma’s, III, Publ. Math. Debrecen 65 (2004), 305–316.CrossRefGoogle Scholar
[Bug08]Bugeaud, Y., Diophantine approximation and Cantor sets, Math. Ann. 341 (2008), 677–684.CrossRefGoogle Scholar
[Bug]Bugeaud, Y., On the rational approximation to the Thue–Morse–Mahler numbers, Ann. Inst. Fourier., to appear.Google Scholar
[Bug10]Bugeaud, Y., Continued fractions of transcendental numbers, Preprint (2010), arXiv:1012.1709.Google Scholar
[BKS11]Bugeaud, Y., Krieger, D. and Shallit, J., Morphic and automatic words: maximal blocks and diophantine approximation, Acta Arith. 149 (2011), 181–199.CrossRefGoogle Scholar
[BL05]Bugeaud, Y. and Laurent, M., Exponents of Diophantine approximation and Sturmian continued fractions, Ann. Inst. Fourier (Grenoble) 55 (2005), 773–804.CrossRefGoogle Scholar
[Cob68]Cobham, A., On the Hartmanis–Stearns problem for a class of tag machines, in Conference record of 1968 ninth annual symposium on switching and automata theory, Schenectady, NY, 1968, 51–60.CrossRefGoogle Scholar
[Cob72]Cobham, A., Uniform tag sequences, Math. Systems Theory 6 (1972), 164–192.CrossRefGoogle Scholar
[DS67]Davenport, H. and Schmidt, W. M., Approximation to real numbers by quadratic irrationals, Acta Arith. 13 (1967), 169–176.CrossRefGoogle Scholar
[Eil74]Eilenberg, S., Automata, languages, and machines, Vol. A, Pure and Applied Mathematics, vol. 58 (Academic Press, New York, 1974).Google Scholar
[Eve96]Evertse, J.-H., An improvement of the quantitative subspace theorem, Compositio Math. 101 (1996), 225–311.Google Scholar
[Kok39]Koksma, J. F., Über die Mahlersche Klasseneinteilung der transzendenten Zahlen und die Approximation komplexer Zahlen durch algebraische Zahlen, Monatsh. Math. Phys. 48 (1939), 176–189.CrossRefGoogle Scholar
[Mah32]Mahler, K., Zur Approximation der Exponentialfunktionen und des Logarithmus. I, II, J. Reine Angew. Math. 166 (1932), 118–150.CrossRefGoogle Scholar
[MH38]Morse, M. and Hedlund, G. A., Symbolic dynamics, Amer. J. Math. 60 (1938), 815–866.CrossRefGoogle Scholar
[MH40]Morse, M. and Hedlund, G. A., Symbolic dynamics II, Amer. J. Math. 62 (1940), 1–42.CrossRefGoogle Scholar
[Mos92]Mossé, B., Puissances de mots et reconnaissabilité des points fixes d’une substitution, Theoret. Comput. Sci. 99 (1992), 327–334.CrossRefGoogle Scholar
[Per29]Perron, O., Die Lehre von den Ketterbrüchen (Teubner, Leipzig, 1929).Google Scholar
[Que98]Queffélec, M., Transcendance des fractions continues de Thue–Morse, J. Number Theory 73 (1998), 201–211.CrossRefGoogle Scholar
[Que00]Queffélec, M., Irrational numbers with automaton-generated continued fraction expansion, in Dynamical systems (Luminy-Marseille, 1998) (World Scientific Publishing, River Edge, NJ, 2000), 190–198.CrossRefGoogle Scholar
[Sch67]Schmidt, W. M., On simultaneous approximations of two algebraic numbers by rationals, Acta Math. 119 (1967), 27–50.CrossRefGoogle Scholar
[Sch71]Schmidt, W. M., Mahler’s T-numbers, in 1969 Number Theory Institute – Proceedings of Symposia in Pure Mathematics, Vol. XX, State Univ. New York, Stony Brook, NY, 1969 (American Mathematical Society, Providence, RI, 1971), 275–286.Google Scholar