On the distribution of powers of real numbers modulo 1

Abstract

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

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