On a lower bound of Hausdorff dimension of weighted singular vectors

Let $w=(w_1,\dots ,w_d)$ be a $d$-tuple of positive real numbers such that ${\sum }_i{w}_i=1$ and $w_1\ge \cdots \ge w_d$. A $d$-dimensional vector $x=(x_1,\dots ,x_d)\in {\mathbb{R}}^d$ is said to be $w$-singular if for every $\epsilon >0$ there exists $T_0>1$ such that for all $T>T_0$ the system of inequalities $$\underset{1\le i\le d}{\max }\mathrm{\mid }q{x}_i-p_i{\mathrm{\mid }}^{\frac{1}{w_i}}<\frac{\epsilon }{T}\text{and}0<q<T$$ have an integer solution $(\mathbf{p},q)=(p_1,\dots ,p_d,q)\in {\mathbb{Z}}^d\times \mathbb{Z}$. We prove that the Hausdorff dimension of the set of $w$-singular vectors in ${\mathbb{R}}^d$ is bounded below by $d-\frac{1}{1+w_1}$. Our result partially extends the previous result of Liao et al. [Hausdorff dimension of weighted singular vectors in ${\mathbb{R}}^2$, J. Eur. Math. Soc. 22 (2020), 833-875].