A Liouville-like approach for the transcendence of some real numbers (Q1876792)

The main result of this paper is the following: Theorem. Let \((u_n)_{n\geq0}\) be a strictly increasing sequence of positive integers. If there exists some \(\varepsilon>0\) such that \(u_{n+1}>(1+\varepsilon)u_n\) for infinitely many \(n\)'s, then the series \(\,f(z)=\sum_{k=0}^{+\infty} z^{u_k}\,\) takes transcendental values in each point \(z=1/\beta\) where \(\beta\) is a Pisot or Salem number. The proof involves a generalization of the Roth-Ridout theorem due to P. Corvaja.