On the fractional parts of certain sequences of $\xi {\alpha }^n$

Assume that $\alpha >1$ is an algebraic number and $\xi \mathrm{\ne }0$ is a real number. We are concerned with the distribution of the fractional parts of the sequence $(\xi {\alpha }^n)$. Under various Diophantine conditions on $\xi$ and $\alpha$, we obtain lower bounds on the number $n$ with $1\le n\le N$ for which the fractional part of the sequence $(\xi {\alpha }^n{)}_{n\ge 1}$ fall into a prescribed region $I\subset [0,1]$, extending several results in the literature. As an application, we show that the Fourier decay rate of some self-similar measures is logarithmic, generalizing a result of Varjú and Yu.