Trancendence measure for continued fractions involving repetitive or symmetric patterns

Transcendence measures for continued fractions involving repetitive or symmetric patterns Boris ADAMCZEWSKI (Lyon) & Yann BUGEAUD (Strasbourg) 1. Introduction 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].

• yields among


• among them


• thue–morse infinite


• numbers shown


• has bounded partial


• transcendental continued