On the distribution of powers of real numbers modulo 1

Given a strictly increasing sequence of positive real numbers tending to infinity $(q_n{)}_{n=1}^{\mathrm{\infty }}$, and an arbitrary sequence of real numbers $(r_n{)}_{n=1}^{\mathrm{\infty }}\mathrm{.}$ We study the set of $\alpha \in (1,\mathrm{\infty })$ for which ${\lim }_{n\to \mathrm{\infty }}\mathrm{\parallel }{\alpha }^{q_n}-r_n\mathrm{\parallel }=0$. In \cite{Dub} Dubickas showed that whenever ${\lim }_{n\to \mathrm{\infty }}(q_{n+1}-q_n)=\mathrm{\infty },$ there always exists a transcendental $\alpha$ for which $\lim_{n\to\infty}|\alpha^{q_{n}}-r_{n}|= 0.$ Adapting the approach of Bugeaud and Moshchevitin \cite{BugMos}, we improve upon this result and show that whenever ${\lim }_{n\to \mathrm{\infty }}(q_{n+1}-q_n)=\mathrm{\infty },$ the set of $\alpha \in (1,\mathrm{\infty })$ satisfying ${\lim }_{n\to \mathrm{\infty }}\mathrm{\parallel }{\alpha }^{q_n}-r_n\mathrm{\parallel }=0$ is a dense set of Hausdorff dimension $1$.