A note on some applications of the subspace theorem

Let f(z)=∑n=0∞anzn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(z) = \sum _{n=0}^{\infty } a_n z^n$$\end{document} be a power series with integer coefficients and converging in the disc D={z:|z|<R}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D = \{ z : |z| < R \}$$\end{document} for some R>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R > 0$$\end{document}. In 1985, Laohakosol proved, using Ridout theorem, that the largest prime factors of partial sums of f(b) for a rational number 0<|b|<R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0< |b| < R$$\end{document} is unbounded, if f(b) is a non-zero algebraic number. In this article, we prove, using the subspace theorem, similar results for other approximation of f(b). Moreover, we prove the number field analogue of Laohakosol’s result.