Automatic continued fractions are transcendental or quadratic

TL;DR

This paper introduces new combinatorial criteria for the transcendence of continued fractions, showing that algebraic numbers of degree at least three have partial quotient sequences with high complexity and cannot be automaton-generated.

Contribution

It provides novel combinatorial conditions that distinguish algebraic numbers of degree at least three from quadratic or rational ones based on their continued fraction partial quotients.

Findings

Sequences of partial quotients for algebraic numbers of degree ≥ 3 are not automaton-generated.

Such sequences cannot have too slow complexity growth.

The criteria help identify transcendental continued fractions.

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.