Let $b\geq2$ be an integer and $A=(a_{ n})_{ n=1}^{ \infty}$ be a strictly increasing subsequence of positive integers with $\eta:=\limsup\limits_{ n\to\infty}\frac{ a_{ n+1}}{ a_{ n}}<+\infty$. 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 $\|b^{ a_n}\xi\|<(b^{ a_N})^{ -\hat{ v}}$ has a solution $n$ with $1\leq n\leq N$. 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 $\hat{ v}_{ b,A}(\xi)\geq\hat{ v}$ ($\hat{ v}_{ b,A}(\xi)=\hat{ v}$) 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 $\hat{ v}$.
Categories: math.NT math.DS