On the algebraicity of Thue–Morse and period-doubling continued fractions
Volume 203 / 2022
Abstract
The link between automaticity and algebraicity is well established concerning power series in finite characteristics, decimal expansion and continued fraction expansion of real numbers. But the question of whether continued fractions (objects in $\mathbb {F}_q[[1/x]]$) defined by automatic sequences taking values in $\mathbb {F}_q[x]$ are algebraic is still wide open and little studied. We approach this problem by investigating the cases of two classical automatic sequences, namely the Thue–Morse and period-doubling sequences. For each sequence, there are infinitely many cases because of the choice of polynomials representing the terms of the sequence. We present our Guess’n’Prove method, which is implemented to give computer-generated proofs of particular instances. We believe our method works for the general case and put forward conjectures.
