Automatic continued fractions are transcendental or quadratic
[Les fractions continues automatiques sont transcendantes ou quadratiques]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 46 (2013) no. 6, pp. 1005-1022.

Nous établissons de nouveaux critères combinatoires de transcendance pour des développements en fraction continue. Soit α=[0;a1,a2,...] un nombre algébrique de degré au moins égal à trois. L’un de nos critères entraîne que la suite (aℓ)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;a1,a2,...] be an algebraic number of degree at least three. One of our criteria implies that the sequence of partial quotients (aℓ)ℓ≥1 of α is not ‘too simple’ (in a suitable sense) and cannot be generated by a finite automaton.

DOI : 10.24033/asens.2208
Classification : 11J70, 11J81, 11J87
Keywords: continued fractions, transcendence
Mot clés : fractions continues, transcendance 
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     title = {Automatic continued fractions are transcendental or quadratic},
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Bugeaud, Yann. Automatic continued fractions are transcendental or quadratic. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 46 (2013) no. 6, pp. 1005-1022. doi : 10.24033/asens.2208. https://www.numdam.org/articles/10.24033/asens.2208/

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