Uniform Diophantine approximation related to $b$-ary and $eta$-expansionsReport as inadecuate

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1 IRMA - Institut de Recherche Mathématique Avancée 2 LAMA - Laboratoire d-Analyse et de Mathématiques Appliquées

Abstract : 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 $\xi$ 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 $T^n \beta \xi$, where $T \beta$ denotes the classical $\beta$-transformation.

Keywords : uniform Diophantine approximation Hausdorff dimension b-ary expansions beta-expansions

Author: Yann Bugeaud - Lingmin Liao -

Source: https://hal.archives-ouvertes.fr/


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