Transcendence of numbers with an expansion in a subclass of complexity 2n + 1

Tomi Kärki

RAIRO - Theoretical Informatics and Applications (2006)

  • Volume: 40, Issue: 3, page 459-471
  • ISSN: 0988-3754

Abstract

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We divide infinite sequences of subword complexity 2n+1 into four subclasses with respect to left and right special elements and examine the structure of the subclasses with the help of Rauzy graphs. Let k ≥ 2 be an integer. If the expansion in base k of a number is an Arnoux-Rauzy word, then it belongs to Subclass I and the number is known to be transcendental. We prove the transcendence of numbers with expansions in the subclasses II and III.

How to cite

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Kärki, Tomi. "Transcendence of numbers with an expansion in a subclass of complexity 2n + 1." RAIRO - Theoretical Informatics and Applications 40.3 (2006): 459-471. <http://eudml.org/doc/249703>.

@article{Kärki2006,
abstract = { We divide infinite sequences of subword complexity 2n+1 into four subclasses with respect to left and right special elements and examine the structure of the subclasses with the help of Rauzy graphs. Let k ≥ 2 be an integer. If the expansion in base k of a number is an Arnoux-Rauzy word, then it belongs to Subclass I and the number is known to be transcendental. We prove the transcendence of numbers with expansions in the subclasses II and III. },
author = {Kärki, Tomi},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Transcendental numbers; subword complexity; Rauzy graph.; transcendental numbers},
language = {eng},
month = {10},
number = {3},
pages = {459-471},
publisher = {EDP Sciences},
title = {Transcendence of numbers with an expansion in a subclass of complexity 2n + 1},
url = {http://eudml.org/doc/249703},
volume = {40},
year = {2006},
}

TY - JOUR
AU - Kärki, Tomi
TI - Transcendence of numbers with an expansion in a subclass of complexity 2n + 1
JO - RAIRO - Theoretical Informatics and Applications
DA - 2006/10//
PB - EDP Sciences
VL - 40
IS - 3
SP - 459
EP - 471
AB - We divide infinite sequences of subword complexity 2n+1 into four subclasses with respect to left and right special elements and examine the structure of the subclasses with the help of Rauzy graphs. Let k ≥ 2 be an integer. If the expansion in base k of a number is an Arnoux-Rauzy word, then it belongs to Subclass I and the number is known to be transcendental. We prove the transcendence of numbers with expansions in the subclasses II and III.
LA - eng
KW - Transcendental numbers; subword complexity; Rauzy graph.; transcendental numbers
UR - http://eudml.org/doc/249703
ER -

References

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  1. B. Adamczewski, Y. Bugeaud and F. Luca, Sur la complexité des nombres algébriques. C. R. Acad. Sci. Paris, Ser. I339 (2004) 11–14.  
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  4. J.-P. Allouche and L.Q. Zamboni, Algebraic irrational binary numbers cannot be fixed points of non-trivial constant length or primitive morphisms. J. Number Theory69 (1998) 119–124.  Zbl0918.11016
  5. P. Arnoux and G. Rauzy, Représentation géométrique de suites de complexité 2n + 1. Bull. Soc. Math. France119 (1991) 199–215.  Zbl0789.28011
  6. S. Ferenczi and C. Mauduit, Transcendence of numbers with a low complexity expansion. J. Number Theory67 (1997) 146–161.  Zbl0895.11029
  7. G.A. Hedlund and M. Morse, Symbolic dynamics II: Sturmian trajectories. Amer. J. Math.62 (1940) 1–42.  Zbl0022.34003
  8. D. Ridout, Rational approximations to algebraic numbers. Mathematika4 (1957) 125–131.  Zbl0079.27401
  9. R.N. Risley and L.Q. Zamboni, A generalization of Sturmian sequences: combinatorial structure and transcendence. Acta Arith.95 (2000), 167–184.  Zbl0953.11007

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