Uniform Diophantine approximation with restricted denominators

Abstract

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