Automatic continued fractions are transcendental or quadratic
[Les fractions continues automatiques sont transcendantes ou quadratiques]
Bugeaud, Yann
Annales scientifiques de l'École Normale Supérieure, Tome 46 (2013), p. 1005-1022 / Harvested from Numdam
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Nous établissons de nouveaux critères combinatoires de transcendance pour des développements en fraction continue. Soit α=[0;a 1 ,a 2 ,...] un nombre algébrique de degré au moins égal à trois. L’un de nos critères entraîne que la suite (a ℓ ) ℓ≥1 des quotients partiels de α n’est pas trop simple (en un certain sens) et ne peut pas être engendrée par un automate fini.

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.

Publié le : 2013-01-01
DOI : https://doi.org/10.24033/asens.2208
Classification:  11J70,  11J81,  11J87
Mots clés: fractions continues, transcendance 
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     author = {Bugeaud, Yann},
     title = {Automatic continued fractions are transcendental or quadratic},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     volume = {46},
     year = {2013},
     pages = {1005-1022},
     doi = {10.24033/asens.2208},
     language = {en},
     url = {http://dml.mathdoc.fr/item/ASENS_2013_4_46_6_1005_0}
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Bugeaud, Yann. Automatic continued fractions are transcendental or quadratic. Annales scientifiques de l'École Normale Supérieure, Tome 46 (2013) pp. 1005-1022. doi : 10.24033/asens.2208. http://gdmltest.u-ga.fr/item/ASENS_2013_4_46_6_1005_0/

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