On the rational approximations to the powers of an algebraic number

Abstract

About fifty years ago Mahler proved that if $\alpha>1$ is rational but not an integer and if $0<l<1$ then the fractional part of $\alpha^n$ is $>l^n$ apart from a finite set of integers $n$ depending on $\alpha$ and $l$. Answering completely a question of Mahler we show that the same conclusion holds for all algebraic numbers which are not $d$-th roots of Pisot numbers. By related methods, we also answer a question by Mendes France, characterizing completely the quadratic irrationals $\alpha$ such that the continued fraction of $\alpha^n$ has period length tending to infinity.