Uniform Diophantine approximation related to $b$-ary and $β$-expansions

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

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