Uniform Diophantine approximation related to -ary and -expansions

Let $b\geq 2$ be an integer and $\hat{v}$ a real number. Among other results, we compute the Hausdorff dimension of the set of real numbers ${\it\xi}$ with the property that, for every sufficiently large integer $N$, there exists an integer $n$ such that $1\leq n\leq N$ and the distance between $b^{n}{\it\xi}$ and its nearest integer is at most equal to $b^{-\hat{v}N}$. We further solve the same question when replacing $b^{n}{\it\xi}$ by $T_{{\it\beta}}^{n}{\it\xi}$, where $T_{{\it\beta}}$ denotes the classical ${\it\beta}$-transformation.