Self-normal numbers

Anne Bertrand-Mathis

Bulletin de la Société Mathématique de France (2013)

  • Volume: 141, Issue: 1, page 25-33
  • ISSN: 0037-9484

Abstract

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Using a method of Champernowne we propose a construction of self-normal numbers in the sense of Schmeling ; these numbers are dense in 1 , ∞ and form a non enumerable set.

How to cite

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Bertrand-Mathis, Anne. "Nombres self normaux." Bulletin de la Société Mathématique de France 141.1 (2013): 25-33. <http://eudml.org/doc/272569>.

@article{Bertrand2013,
abstract = {Nous inspirant de la construction de Champernowne d’un nombre normal en base 10 nous construisons un ensemble de nombres “self-normaux“ au sens de Schmeling ; cet ensemble est non dénombrable et dense dans $[1,\infty [$.},
author = {Bertrand-Mathis, Anne},
journal = {Bulletin de la Société Mathématique de France},
keywords = {normal numbers; generic points; numeration; symbolic dynamics; prefix codes},
language = {fre},
number = {1},
pages = {25-33},
publisher = {Société mathématique de France},
title = {Nombres self normaux},
url = {http://eudml.org/doc/272569},
volume = {141},
year = {2013},
}

TY - JOUR
AU - Bertrand-Mathis, Anne
TI - Nombres self normaux
JO - Bulletin de la Société Mathématique de France
PY - 2013
PB - Société mathématique de France
VL - 141
IS - 1
SP - 25
EP - 33
AB - Nous inspirant de la construction de Champernowne d’un nombre normal en base 10 nous construisons un ensemble de nombres “self-normaux“ au sens de Schmeling ; cet ensemble est non dénombrable et dense dans $[1,\infty [$.
LA - fre
KW - normal numbers; generic points; numeration; symbolic dynamics; prefix codes
UR - http://eudml.org/doc/272569
ER -

References

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  1. [1] B. Adamczewski & Y. Bugeaud – « Dynamics for β -shifts and Diophantine approximation », Ergodic Theory Dynam. Systems27 (2007), p. 1695–1711. Zbl1140.11035MR2371591
  2. [2] A. Bertrand-Mathis – « Développement en base θ ; répartition modulo un de la suite ( x θ n ) n ≥ 0 ; langages codés et θ -shift », Bull. Soc. Math. France114 (1986), p. 271–323. MR878240
  3. [3] —, « Points génériques de Champernowne sur certains systèmes codes ; application aux θ -shifts », Ergodic Theory Dynam. Systems8 (1988), p. 35–51. Zbl0657.28014MR939059
  4. [4] F. Blanchard & G. Hansel – « Systèmes codés », Theoret. Comput. Sci.44 (1986), p. 17–49. MR858689
  5. [5] D. G. Champernowne – « The Construction of Decimals Normal in the Scale of Ten », J. London Math. Soc.8 (1933), p. 254–260. Zbl0007.33701MR1573965
  6. [6] W. Parry – « On the β -expansions of real numbers », Acta Math. Acad. Sci. Hungar.11 (1960), p. 401–416. Zbl0099.28103MR142719
  7. [7] A. Rényi – « Representations for real numbers and their ergodic properties », Acta Math. Acad. Sci. Hungar8 (1957), p. 477–493. Zbl0079.08901MR97374
  8. [8] J. Schmeling – « Symbolic dynamics for β -shifts and self-normal numbers », Ergodic Theory Dynam. Systems17 (1997), p. 675–694. Zbl0908.58017MR1452189

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