The Thue-Morse continued fractions in characteristic 2 are algebraic
A 1979 theorem of \textit{G. Christol} [Theor. Comput. Sci. 9, 141--145 (1979; Zbl 0402.68044)]; also see the 1980 paper of \textit{G. Christol} et al. [Bull. Soc. Math. Fr. 108, 401--419 (1980; Zbl 0472.10035)] states that a formal power series \(\sum_{n \geq 0} a_n X^{-n}\) with coefficients in a field \(K\) of characteristic \(p\) is algebraic over \(K(X)\) if and only if the sequence of its coefficients is \(p\)-automatic (i.e., it satisfies some combinatorial condition that we will not develop here). For such a series the sequence of its partial quotients, in the case where it takes only finitely many values, is sometimes \(p\)-automatic; but this is not always the case: there is no general result on the subject yet. A kind of reverse problem can be studied, namely, given a formal power series (with coefficients in a field \(K\) of characteristic \(p\)) whose sequence of partial quotients is \(p\)-automatic, is it true that the series is algebraic over \(K(X)\)? Again, no general result is known, but partial results are available. In the paper under review, the authors study in detail formal power series \(\xi_{a,b}\) whose partial quotients are two polynomials \(a\) and \(b\) in \({\mathbb F}_2[X]\) alternating like the Thue-Morse sequence. The main result is that \(\xi_{a,b}\) is algebraic of degree \(4\) over \({\mathbb F}_2(X)\) (the authors give an explicit equation satisfied by \(\xi_{a,b}\)). This generalizes a result of \textit{Y. Hu} and the second author [Acta Arith. 203, No. 4, 353--381 (2022; Zbl 1498.11083)] and proves a conjecture in that paper. A corollary is that {\em there are algebraic power series of degree \(4\) in \({\mathbb F}_2((X^{-1}))\) with arbitrarily prescribed Lagrange constants belonging to \(\{2^{-k} | k \geq 1\} \cup \{0\}\)}. Furthermore, the authors show that \(\xi_{a,b}\) satisfies a Ricatti equation but is not hyperquadratic. Also, they compute the higher degree exponents of approximation of \(\xi_{a,b}\). The reader interested in further developments on this theme can consult recent papers see [\textit{Y. Hu}, Finite Fields Appl. 88, Article ID 102191, 17 p. (2023; Zbl 1536.11111); \textit{Y. Hu} and \textit{A. Lasjaunias}, Ann. Inst. Fourier 74, No. 5, 1809--1817 (2024; Zbl 07928996); \textit{Y. Hu}, Int. Math. Res. Not. 2024, No. 9, 7255--7280 (2024; Zbl 07931862)].
scientific article; zbMATH DE number 7903803
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