Hausdorff Dimension of Weighted Singular Vectors

Source abstract

Let $d\ge2$ and let $\mathbf w=(w_1,\ldots,w_d)$ satisfy $w_1\ge\cdots\ge w_d>0$ and $\sum_i w_i=1$. Set $s_*=d-(1+w_1)^{-1}$. We prove that there exist constants $C_{d,\mathbf w}>0$ and $\varepsilon_0=\varepsilon_0(d,\mathbf w)>0$ such that for all $0<\varepsilon<\varepsilon_0$, $$\dim_H\operatorname{DI}_{\mathbf w}(\varepsilon)\le s_*+C_{d,\mathbf w}\sqrt\varepsilon.$$ Together with the lower bound of Kim--Park, this gives the exact formula $$\dim_H\operatorname{Sing}(\mathbf w)=s_*.$$