Automatic continued fractions are transcendental or quadratic

Abstract

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 $(a_{\ell }{)}_{\ell \ge 1}$ of $\alpha$ cannot be generated by a finite automaton, and that the complexity function of $(a_{\ell }{)}_{\ell \ge 1}$ cannot increase too slowly.

20 pages; the title has changed