Automatic continued fractions are transcendental or quadratic

Yann Bugeaud

Annales scientifiques de l'École Normale Supérieure (2013)

  • Volume: 46, Issue: 6, page 1005-1022
  • ISSN: 0012-9593

Abstract

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We establish new combinatorial transcendence criteria for continued fraction expansions. Let  α = [ 0 ; a 1 , a 2 , ... ] be an algebraic number of degree at least three. One of our criteria implies that the sequence of partial quotients ( a ℓ ) ℓ ≥ 1 of  α is not ‘too simple’ (in a suitable sense) and cannot be generated by a finite automaton.

How to cite

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Bugeaud, Yann. "Automatic continued fractions are transcendental or quadratic." Annales scientifiques de l'École Normale Supérieure 46.6 (2013): 1005-1022. <http://eudml.org/doc/272120>.

@article{Bugeaud2013,
abstract = {We establish new combinatorial transcendence criteria for continued fraction expansions. Let $\alpha = [0; a_1, a_2, \ldots ]$ be an algebraic number of degree at least three. One of our criteria implies that the sequence of partial quotients $(a_\{\ell \})_\{\ell \ge 1\}$ of $\alpha $ is not ‘too simple’ (in a suitable sense) and cannot be generated by a finite automaton.},
author = {Bugeaud, Yann},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {continued fractions; transcendence},
language = {eng},
number = {6},
pages = {1005-1022},
publisher = {Société mathématique de France},
title = {Automatic continued fractions are transcendental or quadratic},
url = {http://eudml.org/doc/272120},
volume = {46},
year = {2013},
}

TY - JOUR
AU - Bugeaud, Yann
TI - Automatic continued fractions are transcendental or quadratic
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2013
PB - Société mathématique de France
VL - 46
IS - 6
SP - 1005
EP - 1022
AB - We establish new combinatorial transcendence criteria for continued fraction expansions. Let $\alpha = [0; a_1, a_2, \ldots ]$ be an algebraic number of degree at least three. One of our criteria implies that the sequence of partial quotients $(a_{\ell })_{\ell \ge 1}$ of $\alpha $ is not ‘too simple’ (in a suitable sense) and cannot be generated by a finite automaton.
LA - eng
KW - continued fractions; transcendence
UR - http://eudml.org/doc/272120
ER -

References

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