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Diophantine properties of real numbers generated by finite automata

Published online by Cambridge University Press:  24 November 2006

Boris Adamczewski  and
Julien Cassaigne
Boris Adamczewski
Affiliation:
CNRS, Institut Camille Jordan, Université Claude Bernard Lyon 1, Bât. Braconnier, 21 avenue Claude Bernard, 69622 Villeurbanne cedex, Franceboris.adamczewski@math.univ-lyon1.fr
Julien Cassaigne
Affiliation:
CNRS, Institut de Mathématiques de Luminy, Case 907, 163 avenue de Luminy, 13288 Marseille cedex 9, Francejulien.cassaigne@iml.univ-mrs.fr
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Abstract

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We study some Diophantine properties of automatic real numbers and we present a method to derive irrationality measures for such numbers. As a consequence, we prove that the $b$-adic expansion of a Liouville number cannot be generated by a finite automaton, a conjecture due to Shallit.

Keywords

automatic sequencesLiouville's numbersmeasures of irrationalityMahler's classification11B85 (primary)11J82 (secondary)
Type
Research Article
Information
Compositio Mathematica , Volume 142 , Issue 6 , November 2006 , pp. 1351 - 1372
Copyright
Foundation Compositio Mathematica 2006