Badly approximable points in twisted Diophantine approximation and Hausdorff dimension
Volume 177 / 2017
Abstract
For any $j_1,\ldots,j_n \gt 0$ with $\sum_{i=1}^nj_i=1$ and any $\theta\in\mathbb R^n$, let ${\mathrm{Bad}_{\theta}(j_1,\ldots,j_n)}$ denote the set of points $\eta\in\mathbb R^n$ for which $\max_{1\leq i\leq n}(\|q\theta_i-\eta_i\|^{1/j_i}) \gt c/q$ for some positive constant $c=c(\eta)$ and all $q\in\mathbb N$. These sets are the ‘twisted’ inhomogeneous analogue of $\mathrm{Bad}(j_1,\ldots,j_n)$ in the theory of simultaneous Diophantine approximation. It has been shown that they have full Hausdorff dimension in the non-weighted setting, i.e. provided that $j_i=1/n$, and in the weighted setting when $\theta$ is chosen from $\mathrm{Bad}(j_1,\ldots,j_n)$. We generalise these results by proving the full Hausdorff dimension in the weighted setting without any condition on $\theta$. Moreover, we prove $\dim(\mathrm{Bad}_{\theta}(j_1,\ldots,j_n)\cap\mathrm{Bad}(1,0,\ldots,0)\cap\cdots\cap\mathrm{Bad}(0,\ldots,0,1))=n$.
