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arXiv · 1012.1709

Automatic continued fractions are transcendental or quadratic

Yann Bugeaud
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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.

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Yann Bugeaud. 2010-12-08. Automatic continued fractions are transcendental or quadratic. https://arxiv.org/abs/1012.1709

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