Badly approximable points in twisted Diophantine approximation and Hausdorff dimension
Tom 177 / 2017
Streszczenie
For any 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.