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

Published online by Cambridge University Press:  04 August 2014

Université de Strasbourg, IRMA, 7, rue René Descartes, 67084 Strasbourg, France email
Université Paris-Est Créteil, LAMA 61, av Général de Gaulle, 94000 Créteil, France email


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.

Research Article
© Cambridge University Press, 2014 

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