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Irrationality measures for some automatic real numbers

Published online by Cambridge University Press:  10 June 2009

BORIS ADAMCZEWSKI  and
TANGUY RIVOAL
BORIS ADAMCZEWSKI
Affiliation:
CNRS, Université de Lyon, Université Lyon 1, Institut Camille Jordan, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France. e-mail: [email protected]
TANGUY RIVOAL
Affiliation:
CNRS, Université Grenoble I, Institut Fourier, 100 Rue des Maths, BP 74, 38402 Saint-Martin-d'Hères cedex, France. e-mail: [email protected]
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Abstract

This paper is devoted to the rational approximation of automatic real numbers, that is, real numbers whose expansion in an integer base can be generated by a finite automaton. We derive upper bounds for the irrationality exponent of famous automatic real numbers associated with the Thue–Morse, Rudin–Shapiro, paperfolding and Baum–Sweet sequences. These upper bounds arise from the construction of some explicit Padé or Padé type approximants for the generating functions of these sequences. In particular, we prove that the Thue–Morse–Mahler numbers have an irrationality exponent at most equal to 4. We also obtain an explicit description of infinitely many convergents to these numbers.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 147 , Issue 3 , November 2009 , pp. 659 - 678
Copyright
Copyright © Cambridge Philosophical Society 2009

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References

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