We 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+ (NnaM mod 2)) n \ 0 , or are a ''repetitive'' fixed point of a binary morphism satisfying some technical conditions.

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Abstract.We discuss certain simple continued fractions that exhibit a type of "self-similar" structure: their partial quotients are formed by perturbing and shifting the denominators of their convergents. We prove that all such continued fractions represent transcendental numbers. As an application, we prove that Cahen's constantis transcendental. Here (Sn) is Sylvester's sequence defined by S O = 2 and S, + 1 = $2 --Sn + 1 for n/> 0. We also explicitly compute the continued fraction for the number C; its partial quotients grow doubly exponentially and they are all squares.

Abstract. Let a = (1 + y5)/2. In this paper it is proved that GO

If α is not a quadratic irrational, then we produce a specific sequence of quadratic irrational approximations to α, the rate of convergence given in terms of L and γ. As an application, we demonstrate the transcendence of some continued fractions, a typical one being of the form [0, u 1 , u 2 , . . . ] with um = 1 + mθ mod n, n 2, and θ an irrational number which satisfies any of a given set of conditions.