Hausdorff Dimension in Inhomogeneous Diophantine Approximation

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.