This interesting article is nicely described by (selected) parts of the introduction: ``About fifty years ago \textit{K. Mahler} [Mathematika 4, 122--124 (1957; Zbl 0208.31002)] proved that if \(\alpha> 1\) is rational but not an integer and if \(01\) all of whose conjugates different from \(\alpha\) have absolute value less than 1. Mahler also added that ``It would be of some interest to know which algebraic numbers have the same property as (the rationals in the theorem)''. Now, it seems that even replacing Ridout's theorem with the modern versions of Roth's theorem, valid for several valuations and approximations in any given number field, the method of Mahler does not lead to a complete solution to his question. One of the objects of the present paper is to answer Mahler's question completely; our methods will involve a suitable version of the Schmidt subspace theorem, which may be considered as a multi-dimensional extension of the results mentioned by Roth, Mahler and Ridout. We state at once our first theorem, where as usual we denote by \(\| x\|\) the distance of the complex number \(x\) from the nearest integer in \(\mathbb Z\), i.e. \(\| x\| := \min\{| x-m| : m\in\mathbb Z\}\). Theorem 1. Let \(\alpha>1\) be a real algebraic number and let \(00\) be a real quadratic irrational. If \(\alpha\) is neither the square root of a rational number, nor a unit in the ring of integers of \(\mathbb Q(\alpha)\), then the period length of the continued fraction for \(\alpha^n\) tends to infinity with \(n\). If \(\alpha\) is the square root of a rational number, the period length of the continued fraction for \(\alpha^{2n+1}\) tends to infinity. If \(\alpha\) is a unit, the period length of the continued fraction for \(\alpha^n\) is bounded. If \(\alpha\) is the square root of a rational number, then the continued fraction for \(\alpha^n\) is finite, so Theorem 2 gives a complete answer to a problem posed by Mendès France. The main tool in the proof of both theorems is the following new lower bound for the fractional parts of \(S\)-units in algebraic number fields. Definition. We call a (complex) algebraic number \(\alpha\) a pseudo-Pisot number if (i) \(|\alpha | > 1\) and all its conjugates have (complex) absolute value strictly less than 1; (ii) \(\alpha\) has an integral trace: \(\text{Tr}_{\mathbb Q(\alpha)/\mathbb Q}(\alpha)\in\mathbb Z\). Of course, pseudo-Pisot numbers are ``well approximated'' by their trace, hence are good candidates for having a small fractional part compared to their height. The algebraic integers among the pseudo-Pisot numbers are just the usual Pisot numbers. We prove the following result: Main Theorem. Let \(\Gamma\subset\overline{\mathbb Q}^\times\) be a finitely generated multiplicative group of algebraic numbers, let \(\delta\in\overline{\mathbb Q}^\times\) be a non-zero algebraic number and let \(\varepsilon>0\) be fixed. Then there are only finitely many pairs \((q,u)\in\mathbb Z\times\Gamma\) with \(d=[\mathbb Q(u):\mathbb Q]\) such that \(|\delta qu|>1\), \(\delta qu\) is not a pseudo-Pisot number and \[ 0 0\). The above main theorem can be viewed as a Thue-Roth inequality with ``moving target'', as the theorem in the authors' paper [J. Théor. Nombres Bordx. 17, No. 3, 737--748 (2005; Zbl 1159.11021)], where we considered quotients of power sums with integral roots instead of elements of a finitely generated multiplicative group. The main application of the theorem in the paper cited above also concerned continued fractions, as for our Theorem 2.''