Transcendence of Sturmian or morphic continued fractions (Q1612522)

The authors prove, using a theorem of W. Schmidt, that if the sequence of partial quotients of the continued fraction expansion of a positive irrational real number takes only two values and begins with arbitrary long blocks which are `almost squares', then this number is either quadratic or transcendental. This result applies in particular to real numbers whose partial quotients form a Sturmian (or quasi-Sturmian) sequence, or are given by the sequence \((1 + (\lfloor n\alpha \rfloor \bmod 2))_{n\geq 0}\), or are a `repetitive' fixed point of a binary morphism satisfying some technical conditions.