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 $w_1\geq \cdots \geq w_d$. 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 \[ \max_{1\leq i\leq d}|qx_i - p_i|^{\frac{1}{w_i}} < \frac{\epsilon}{T} \quad\text{and}\quad 0<q<T \] 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].