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Badly approximable points in twisted Diophantine approximation and Hausdorff dimension

Badly approximable points in twisted Diophantine approximation and Hausdorff dimension

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