Non literal tranducers and some problems of normality

François Blanchard

Journal de théorie des nombres de Bordeaux (1993)

  • Volume: 5, Issue: 2, page 303-321
  • ISSN: 1246-7405

Abstract

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A new proof of Maxfield’s theorem is given, using automata and results from Symbolic Dynamics. These techniques permit to prove that points that are near normality to base p k (resp. p ) are also near normality to base p (resp. p k ), and to study genericity preservation for non Lebesgue measures when going from one base to the other. Finally, similar results are proved to bases the golden mean and its square.

How to cite

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Blanchard, François. "Non literal tranducers and some problems of normality." Journal de théorie des nombres de Bordeaux 5.2 (1993): 303-321. <http://eudml.org/doc/93584>.

@article{Blanchard1993,
abstract = {A new proof of Maxfield’s theorem is given, using automata and results from Symbolic Dynamics. These techniques permit to prove that points that are near normality to base $p^k$ (resp. $p$) are also near normality to base $p$ (resp. $p^k$), and to study genericity preservation for non Lebesgue measures when going from one base to the other. Finally, similar results are proved to bases the golden mean and its square.},
author = {Blanchard, François},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {transducers; normality; real numbers; base changes; automata; Maxfield's theorem},
language = {eng},
number = {2},
pages = {303-321},
publisher = {Université Bordeaux I},
title = {Non literal tranducers and some problems of normality},
url = {http://eudml.org/doc/93584},
volume = {5},
year = {1993},
}

TY - JOUR
AU - Blanchard, François
TI - Non literal tranducers and some problems of normality
JO - Journal de théorie des nombres de Bordeaux
PY - 1993
PB - Université Bordeaux I
VL - 5
IS - 2
SP - 303
EP - 321
AB - A new proof of Maxfield’s theorem is given, using automata and results from Symbolic Dynamics. These techniques permit to prove that points that are near normality to base $p^k$ (resp. $p$) are also near normality to base $p$ (resp. $p^k$), and to study genericity preservation for non Lebesgue measures when going from one base to the other. Finally, similar results are proved to bases the golden mean and its square.
LA - eng
KW - transducers; normality; real numbers; base changes; automata; Maxfield's theorem
UR - http://eudml.org/doc/93584
ER -

References

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  5. [BIDT] F. Blanchard, J.-M. Dumont, A. Thomas, Generic sequences, transducers and multiplication of normal numbers, Israel J. Math.80 (1992), 257-287. Zbl0776.11040MR1202572
  6. [BrL] A. Broglio, P. Liardet, Predictions with automata, Symbolic Dynamics and its Applications, AMS, providence, RI, P. Walters ed., Contemporary Math135, 1992. Zbl0777.11028MR1185084
  7. [C] J.W.S. Cassels, On a paper of Niven and Zuckerman, Pacific J. Math.3 (1953), 555-557. Zbl0047.04402MR51271
  8. [D] J.-M. Dumont, Private communication. 
  9. [F1] C. Frougny, Representations of numbers and finite automata, Math. Systems Theory25 (1992), 37-60. Zbl0776.11005MR1139094
  10. [F2] C. Frougny, How to write integers in non-integer base, Lecture Notes in Computer Science (Proceedings of Latin '92) 583, 154-164. MR1253354
  11. [F3] C. Frougny, B. Solomyak, Finite Beta-expansions, Ergod. Th. & Dynam. Syst. 12 (1992), 713-723. Zbl0814.68065MR1200339
  12. [K] T. Kamae, Subsequences of normal sequences, Israel J. Math.16 (1973), 121-149. Zbl0272.28012MR338321
  13. [M] J.E. Maxfield, Normal k-tuples, Pacif. J. Math.3 (1953), 189-196. Zbl0050.27503MR53978
  14. [PS] K. Petersen, L. Shapiro, Induced flows, Trans. Amer. Math. Soc.177 (1973), 375-390. Zbl0229.54036MR322839

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