Transcendence of Thue-Morse continued fractions
For distinct positive integers \(a\) and \(b\), the Thue-Morse sequence \(a_1a_2a_3\dots\) is defined to be the fixed point of the substitution \(\zeta(a)=ab\), \(\zeta(b)=ba\) on \(\{a,b\}\); it begins with \[ abbabaabbaababbabaa babbaabba \dots \] Theorem: The real number \(x\) between 0 and 1 with simple continued fraction expansion \(x=[0,a_1,a_2,a_3, \dots]\) is transcendent. The author deduces this result from a theorem due to \textit{W. Schmidt} [Acta Math. 119, 27-50 (1967; Zbl 0173.04801)] which tells us that a real number is transcendent if it is well approximable, in a specified sense, by quadratic irrationals. It is shown that various other criteria for transcendence cannot be used to establish the result.
scientific article; zbMATH DE number 1247919
scientific article