Uniform Diophantine approximation related to $b$-ary and $\beta$-expansions

Abstract

Let $b\ge 2$ be an integer and $\hv$ a real number. Among other results, we compute the Hausdorff dimension of the set of real numbers $\xi$ 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 $b^n\xi$ and its nearest integer is at most equal to $b^{-\hv N}$. We further solve the same question when replacing $b^n\xi$ by $T_{\beta }^n\xi$, where $T_{\beta }$ denotes the classical $\beta$-transformation.

25 pages

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