Transcendence measures for continued fractions involving repetitive or symmetric patterns

It was observed long ago (see e.g., [32] or [20], page 62) that Roth’s theorem [28] and its p-adic extension established by Ridout [27] can be used to prove the transcendence of real numbers whose expansion in some integer base contains repetitive patterns. This was properly written only in 1997, by Ferenczi and Mauduit [21], who adopted a point of view from combinatorics on words before applying the above mentioned theorems from Diophantine approximation to establish e.g., the transcendence of numbers with a low complexity expansion. Their combinatorial transcendence criterion was subsequently considerably improved in [9] by means of the multidimensional extension of Roth’s theorem established by W. M. Schmidt, commonly referred to as the Schmidt Subspace Theorem [29, 30]. As shown in [4], this powerful criterion has many applications and yields among other things the transcendence of irrational real numbers whose expansion in some integer base can be generated by a finite automaton. The latter result was generalized in [7], where we gave transcendence measures for a large class of real numbers shown to be transcendental in [4]. The key ingredient for the proof is then the Quantitative Subspace Theorem [31]. We described in [7] a general method that allows us in principle to get transcendence measures for real numbers that are proved to be transcendental by an application of Roth’s or Schmidt’s theorem, or one of their extensions. Besides expansions in integer bases, a classical way to represent a real number is by its continued fraction expansion. There is actually a long tradition in constructing explicit classes of transcendental continued fractions and especially transcendental continued fractions with bounded partial quotients [24, 13, 26, 11]. Again by means of the Schmidt Subspace Theorem, existing results were recently substantially improved in a series of papers [1, 5, 6, 8, 17], providing new classes of transcendental continued fractions. It is the purpose of the present work to show how the Quantitative Subspace Theorem yields transcendence measures for (most of) these numbers, following the approach initiated in [7]. These measures allow us to locate such numbers in the classification of real numbers defined in 1932 by Mahler [23] and recalled below. For every integer d ≥ 1 and every real number ξ, we denote by wd(ξ) the supremum of the exponents w for which 0 < |P (ξ)| < H(P )−w