Uniform Diophantine approximation with restricted denominators

Abstract

Let $b\ge 2$ be an integer and $A={\left(a_n\right)}_{n=1}^{\infty }$ be a strictly increasing subsequence of positive integers with $\eta :=\underset{n\to \infty }{\mathrm{lim\hspace{0.17em}sup}}\frac{a_{n+1}}{a_n}<+\infty$. For each irrational real number ξ, we denote by ${\hat{v}}_{b,A}\left(\xi \right)$ the supremum of the real numbers $\hat{v}$ for which, for every sufficiently large integer N, the equation $\Vert b^{a_n}\xi \Vert <{\left(b^{a_N}\right)}^{-\hat{v}}$ has a solution n with $1\le n\le N$. For every $\hat{v}\in [0,\eta ]$, let ${\hat{V}}_{b,A}\left(\hat{v}\right)$ (${\hat{V}}_{b,A}^{⁎}\left(\hat{v}\right)$) be the set of all real numbers ξ such that ${\hat{v}}_{b,A}\left(\xi \right)\ge \hat{v}$ (${\hat{v}}_{b,A}\left(\xi \right)=\hat{v}$) respectively. In this paper, we give some results of the Hausdorfff dimensions of ${\hat{V}}_{b,A}\left(\hat{v}\right)$ and ${\hat{V}}_{b,A}^{⁎}\left(\hat{v}\right)$. When $\eta =1$, we prove that the Hausdorfff dimensions of ${\hat{V}}_{b,A}\left(\hat{v}\right)$ and ${\hat{V}}_{b,A}^{⁎}\left(\hat{v}\right)$ are equal to ${\left(\frac{1-\hat{v}}{1+\hat{v}}\right)}^2$ for any $\hat{v}\in [0,1]$. When $\eta >1$ and ${\lim }_{n\to \infty }\frac{a_{n+1}}{a_n}$ exists, we show that the Hausdorfff dimension of ${\hat{V}}_{b,A}\left(\hat{v}\right)$ is strictly less than ${\left(\frac{\eta -\hat{v}}{\eta +\hat{v}}\right)}^2$ for some $\hat{v}$, which is different with the case $\eta =1$, and we give a lower bound of the Hausdorfff dimensions of ${\hat{V}}_{b,A}\left(\hat{v}\right)$ and ${\hat{V}}_{b,A}^{⁎}\left(\hat{v}\right)$ for any $\hat{v}\in [0,\eta ]$. Furthermore, we show that this lower bound can be reached for some $\hat{v}$.