Transcendence of certain k-ary continued fraction expansions

Let ξ ∈ (0, 1) be an irrational with aperiodic continued fraction expansion: ξ = [0; u0, u1, u2, . . .], and suppose the sequence (un)n≥0 of partial quotients takes only values from the finite set {a1, a2, . . . , ak} with 1 ≤ a1 < a2 < · · · < ak, k ≥ 2. We prove that if the frequency of a1 (or ak) in (un)n≥0 is at least 1/2, and (un)n≥0 begins with arbitrarily long blocks that are almost squares, then ξ is transcendental. This extends a result of Allouche et al (2001), who studied the binary case k = 2. Consideration is also given to some examples of such transcendental continued fractions [0; u0, u1, u2, . . .], including the case when (un)n≥0 is an aperiodic strict standard episturmian word, or a certain generalised Thue-Morse word.