There are infinitely many limit points of the fractional parts of powers

Abstract

Suppose that $\al>1$ is an algebraic number and $ξ>0$ is a real number. We prove that the sequence of fractional parts $\{ξ\al^n\},$ $n =1,2,3,...,$ has infinitely many limit points except when $\al$ is a PV-number and $ξ\in \Q(\al).$ For $ξ=1$ and $\al$ being a rational non-integer number, this result was proved by Vijayaraghavan.