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VARIATIONS AROUND A PROBLEM OF MAHLER AND MENDÈS FRANCE

Published online by Cambridge University Press:  25 April 2012

YANN BUGEAUD*
Affiliation:
Université de Strasbourg, Mathématiques, 7, rue René Descartes, 67084 Strasbourg, France (email: bugeaud@math.unistra.fr)
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Abstract

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We discuss the following general question and some of its extensions. Let (εk)k≥1 be a sequence with values in {0,1}, which is not ultimately periodic. Define ξ:=∑ k≥1εk/2k and ξ′:=∑ k≥1εk/3k. Let 𝒫 be a property valid for almost all real numbers. Is it true that at least one among ξ and ξ′ satisfies 𝒫?

Type
Research Article
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

References

[1]Adamczewski, B., ‘On the expansion of some exponential periods in an integer base’, Math. Ann. 346 (2010), 107116.CrossRefGoogle Scholar
[2]Adamczewski, B. and Bugeaud, Y., ‘On the complexity of algebraic numbers I. Expansions in integer bases’, Ann. of Math. (2) 165 (2007), 547565.CrossRefGoogle Scholar
[3]Adamczewski, B. and Rivoal, T., ‘Irrationality measures for some automatic real numbers’, Math. Proc. Cambridge Philos. Soc. 147 (2009), 659678.CrossRefGoogle Scholar
[4]Allouche, J.-P. and Shallit, J. O., Automatic Sequences: Theory, Applications, Generalizations (Cambridge University Press, Cambridge, 2003).CrossRefGoogle Scholar
[5]Amou, M. and Bugeaud, Y., ‘Expansions in integer bases and exponents of Diophantine approximation’, J. Lond. Math. Soc. 81 (2010), 297316.CrossRefGoogle Scholar
[6]Bugeaud, Y., ‘On the rational approximation to the Thue–Morse–Mahler numbers’, Ann. Inst. Fourier (Grenoble) 61 (2011), 16671678.CrossRefGoogle Scholar
[7]Bugeaud, Y., ‘Diophantine approximation and Cantor sets’, Math. Ann. 341 (2008), 677684.CrossRefGoogle Scholar
[8]Bugeaud, Y. and Evertse, J.-H., ‘On two notions of complexity of algebraic numbers’, Acta Arith. 133 (2008), 221250.CrossRefGoogle Scholar
[9]Bugeaud, Y., Krieger, D. and Shallit, J., ‘Morphic and automatic words: maximal blocks and diophantine approximation’, Acta Arith. 149 (2011), 181199.CrossRefGoogle Scholar
[10]Kleinbock, D., Lindenstrauss, E. and Weiss, B., ‘On fractal measures and Diophantine approximation’, Selecta Math. 10 (2004), 479523.CrossRefGoogle Scholar
[11]Kmošek, M., ‘Rozwiniecie niektórych liczb niewymiernych na ułamki łańcuchowe’, Master Thesis, Warsaw, 1979.Google Scholar
[12]Köhler, G., ‘Some more predictable continued fractions’, Monatsh. Math. 89 (1980), 95100.CrossRefGoogle Scholar
[13]Mahler, K., ‘Some suggestions for further research’, Bull. Aust. Math. Soc. 29 (1984), 101108.CrossRefGoogle Scholar
[14]Mendès France, M., ‘Sur les fractions continues limitées’, Acta Arith. 23 (1973), 207215.CrossRefGoogle Scholar
[15]Mendès France, M., ‘Sur les décimales des nombres algébriques réels’, in: Seminar on Number Theory, 1979–1980, Exp. No, 28 (Univ. Bordeaux I, Talence, 1980), p. 7.Google Scholar
[16]Nishioka, K., Mahler Functions and Transcendence, Lecture Notes in Mathematics, 1631 (Springer, Berlin, 1996).CrossRefGoogle Scholar
[17]Pethő, A., ‘Simple continued fractions for the Fredholm numbers’, J. Number Theory 14 (1982), 232236.CrossRefGoogle Scholar
[18]Riyapan, P., Laohakosol, V. and Chaichana, T., ‘Two types of explicit continued fractions’, Period. Math. Hungar. 52 (2006), 5172.CrossRefGoogle Scholar
[19]Shallit, J. O., ‘Simple continued fractions for some irrational numbers’, J. Number Theory 11 (1979), 209217.CrossRefGoogle Scholar
[20]Shallit, J. O., ‘Simple continued fractions for some irrational numbers, II’, J. Number Theory 14 (1982), 228231.CrossRefGoogle Scholar
[21]van der Poorten, A. and Shallit, J. O., ‘Folded continued fractions’, J. Number Theory 40 (1992), 237250.CrossRefGoogle Scholar