Hausdorff dimension in inhomogeneous Diophantine approximation

Let $\alpha$ be an irrational real number. We show that the set of $ϵ$-badly approximable numbers $${\mathrm{B}\mathrm{a}\mathrm{d}}^{\epsilon }(\alpha ):=\{x\in [0,1]:\underset{\mathrm{\mid }q\mathrm{\mid }\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\mathrm{\mid }q\mathrm{\mid }\cdot \mathrm{\parallel }q\alpha -x\mathrm{\parallel }\ge \epsilon \}$$ has full Hausdorff dimension for some positive $ϵ$ 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.