On a lower bound of Hausdorff dimension of weighted singular vectors

Let w=(w1,⋯,wd)$w=(w_1,\dots,w_d)$ be a d$d$ ‐tuple of positive real numbers such that ∑iwi=1$\sum _{i}w_i =1$ and w1⩾⋯⩾wd$w_1\geqslant \cdots \geqslant w_d$ . A d$d$ ‐dimensional vector x=(x1,⋯,xd)∈Rd$x=(x_1,\dots,x_d)\in \mathbb {R}^d$ is said to be w$w$ ‐singular if for every ε>0$\epsilon >0$ , there exists T0>1$T_0>1$ such that for all T>T0$T>T_0$ , the system of inequalities max1⩽i⩽d|qxi−pi|1wi