On integer and fractional parts of powers of Salem numbers

Let N be a positive rational integer and let P be the set of powers of a Salem number of degree d. We prove that for any α∈P the fractional parts of the numbers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{\alpha ^n }}{N}$$\end{document}, when n runs through the set of positive rational integers, are dense in the unit interval if and only if N≦ 2d − 4. We also show that for any α∈P the integer parts of the numbers αn are divisible by N for infinitely many n if and only if N≦ 2d − 3.