We establish new combinatorial transcendence criteria for continued fraction expansions. Let be an algebraic number of degree at least three. One of our criteria implies that the sequence of partial quotients of is not ‘too simple’ (in a suitable sense) and cannot be generated by a finite automaton.
Skolem conjectured that the "power sum" A(n) = λ₁α₁ⁿ + ⋯ + λₘαₘⁿ satisfies a certain local-global principle. We prove this conjecture in the case when the multiplicative group generated by α₁,...,αₘ is of rank 1.
During the last decade, several quite unexpected applications of the Schmidt Subspace Theorem were found. We survey some of these, with a special emphasize on the consequences of quantitative statements of this theorem, in particular regarding transcendence questions.