Uniform Diophantine approximation with restricted denominators
Let $b\geq2$ be an integer and $A=(a_{n})_{n=1}^{\infty}$ be a strictly increasing subsequence of positive integers with $η:=\limsup\limits_{n\to\infty}\frac{a_{n+1}}{a_{n}}<+\infty$. For each irrational real number $ξ$, we denote by $\hat{v}_{b,A}(ξ)$ the supremum of the real numbers $\hat{v}$ for which, for every sufficiently large integer $N$, the equation $\|b^{a_n}ξ\|<(b^{a_N})^{-\hat{v}}$ has a solution $n$ with $1\leq n\leq N$. For every $\hat{v}\in[0,η]$, let $\hat{\mathcal{V}}_{b,A}(\hat{v})$ ($\hat{\mathcal{V}}_{b,A}^{\ast}(\hat{v})$) be the set of all real numbers $ξ$ such that $\hat{v}_{b,A}(ξ)\geq\hat{v}$ ($\hat{v}_{b,A}(ξ)=\hat{v}$) 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 $η=1$, 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 $η>1$ 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 $\left(\frac{η-\hat{v}}{η+\hat{v}}\right)^{2}$ for some $\hat{v}$, which is different with the case $η=1$, 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 $\hat{v}\in[0,η]$. Furthermore, we show that this lower bound can be reached for some $\hat{v}$.