Hausdorff Dimension in Inhomogeneous Diophantine Approximation. (4th April 2019)

• Record Type: Journal Article
• Authors: Bugeaud, Yann; Kim, Dong Han; Lim, Seonhee; Rams, Michał
• Is Part Of: International mathematics research notices. Volume 2021:Number 3(2021)
• Page Start: 2108
• Page End: 2133
• Publication Date: 2019-04-04
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnz073
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: Let $\alpha $ be an irrational real number. We show that the set of $\varepsilon $ -badly approximable numbers $$\begin{equation*} \textrm{Bad}^\varepsilon (\alpha):= \Big\{x\in [0, 1]\, : \, \liminf_{|q| \to \infty} |q| \cdot \| q\alpha -x \| \geq \varepsilon \Big\} \end{equation*}$$ has full Hausdorff dimension for some positive $\varepsilon $ if and only if $\alpha $ is singular on average. The condition is equivalent to the average $\frac{1}{k} \sum _{i=1, \cdots, k} \log a_i$ of the logarithms of the partial quotients $a_i$ of $\alpha $ going to infinity with $k$ . We also consider one-sided approximation, obtain a stronger result when $a_i$ tends to infinity, and establish a partial result in higher dimensions.