Automatic continued fractions are transcendental or quadratic

Bugeaud, Yann

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 $\alpha =[0;a_1,a_2,...]$ un nombre algébrique de degré au moins égal à trois. L’un de nos critères entraîne que la suite ${\left(a_{\ell }\right)}_{\ell \ge 1}$ des quotients partiels de $\alpha$ 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 $\alpha =[0;a_1,a_2,...]$ be an algebraic number of degree at least three. One of our criteria implies that the sequence of partial quotients ${\left(a_{\ell }\right)}_{\ell \ge 1}$ of $\alpha$ is not ‘too simple’ (in a suitable sense) and cannot be generated by a finite automaton.

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