Uniform Diophantine approximation related to beta-transformations

Abstract

For any $\beta >1$, let $T_{\beta }$ be the classical $\beta$-transformations. Fix $x_0\in [0,1]$ and a nonnegative real number $\hat{v}$, we compute the Hausdorff dimension of the set of real numbers $x\in [0,1]$ with the property that, for every sufficiently large integer $N$, there is an integer $n$ with $1\le n\le N$ such that the distance between $T_{\beta }^{n}x$ and $x_0$ is at most equal to ${\beta }^{-N\hat{v}}$. This work extends the result of Bugeaud and Liao \cite{YLiao2016} to every point $x_0$ in unit interval.

arXiv admin note: substantial text overlap with arXiv:1907.13031