Extremal values of continuants and transcendence of certain continued fractions

Maximal continuants and the Fine–Wilf theorem

“…(ii) The complete information about each word A max ( , m) and the path connecting it with its root word is encoded in the generalized continued fraction S( , m) (2). In particular, the unique root ( , m ) associated with a given pair ( , m), and also the nodes and edges of the path connecting both points can be computed explicitly by the expansion (see (9), (11) and (13) below).…”

“…Recently there has been a revival of interest in the topic, starting from an observation by Baxa [2] who used the structural theorem about q max as a crucial tool for establishing the transcendence of the regular continued fraction [a 1 , a 2 , . .…”

Continued fractions and transcendental numbers

“…With the same auxiliary tool, M. Queffélec [32] established the nice result that the Thue-Morse continued fraction is transcendental (see Section 5). This method has then been made more explicit, and combinatorial transcendence criteria based on Davison's approach were given in [7,17,13,25].…”

“…The main novelty in their approach is the use of a stronger Diophantine result of W. M. Schmidt [37,38], commonly known as the Subspace Theorem. After some work, it yields considerable improvements upon the criteria from [7,17,13,25]. These allowed them to prove that the continued fraction expansion of every real algebraic number of degree at least three cannot be "too simple", in various senses.…”

“…In the subsequent Sections, we give various examples of applications of these transcendence criteria. We first solve in Section 4 a problem originally tackled by Davison in [16] and later considered by several authors in [7,17,13]. The Rudin-Shapiro and the Baum-Sweet continued fractions are proved to be transcendental in Section 5.…”