Hostname: page-component-7c8c6479df-24hb2 Total loading time: 0 Render date: 2024-03-28T22:33:31.376Z Has data issue: false hasContentIssue false

Irrationality measures for some automatic real numbers

Published online by Cambridge University Press:  10 June 2009

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: Boris.Adamczewki@math.univ-lyon1.fr
TANGUY RIVOAL
Affiliation:
CNRS, Université Grenoble I, Institut Fourier, 100 Rue des Maths, BP 74, 38402 Saint-Martin-d'Hères cedex, France. e-mail: rivoal@ujf-grenoble.fr

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
Copyright
Copyright © Cambridge Philosophical Society 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Adamczewski, B. and Bugeaud, Y.On the complexity of algebraic numbers I. Expansions in integer bases. Ann. Math. 165 (2007), 547565.Google Scholar
[2]Adamczewski, B. and Bugeaud, Y. Nombres réels de complexité sous-linaire: mesures d'irrationalité et de transcendance, Preprint (2008).Google Scholar
[3]Adamczewski, B., Bugeaud, Y. and Luca, F.Sur la complexité des nombres algébriques, C. R. Acad. Sci. Paris 339 (2004), 1114.CrossRefGoogle Scholar
[4]Adamczewski, B. and Cassaigne, J.Diophantine properties of real numbers generated by finite automata. Comp. Math. 142 (2006), 13511372.Google Scholar
[5]Allouche, J.-P. and Shallit, J. O.Automatic Sequences: Theory, Applications, Generalizations (Cambridge University Press, 2003).Google Scholar
[6]Bugeaud, Y.Diophantine approximation and Cantor sets. Math. Ann. 341 (2008), 677684.Google Scholar
[7]Cobham, A.Uniform tag sequences. Math. Systems Theory 6 (1972), 164192.Google Scholar
[8]Christol, G.Ensembles presques périodiques k-reconnaissables. Theoret. Comp. Sci. 9 (1979), 141145.Google Scholar
[9]Baker, G. A. and Graves–Morris, P.Padé approximants, Second edition. Encyclopedia of Mathematics and its Applications 59 (Cambridge University Press, 1996).Google Scholar
[10]Loxton, J. H. and Poorten, A. J. van derArithmetic properties of automata: regular sequences. J. Reine Angew. Math. 392 (1988), 5769.Google Scholar
[11]Kempner, A. J.On transcendental numbers. Trans. Amer. Math. Soc. 17 (1916), 476482.Google Scholar
[12]Lehr, S.Sums and rational multiples of q-automatic sequences are q-automatic. Theoret. Comp. Sci. 108 (1993), 385391.Google Scholar
[13]Lothaire, M.Algebraic Combinatorics on Words, Vol. 90 of Encyclopedia of Mathematics and Its Applications (Cambridge University Press, 2002).Google Scholar
[14]Mahler, K.Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen. Math. Ann. 101 (1929), 342366. Corrigendum 103 (1930), 532.Google Scholar
[15]Rivoal, T.Convergents and irrationality measures of logarithms. Rev. Mat. Iberoamericana 23.3 (2007), 931952.Google Scholar
[16]Shallit, J. O.Number theory and formal languages, in Emerging applications of number theory, IMA Volumes (Springer, Berlin, 1999), 547570.Google Scholar