AI 中文总结
本文证明了加权奇异向量集合的豪斯多夫维数上界，结合已知下界，得到精确维数公式$s_*=d-(1+w_1)^{-1}$。
AI 中文摘要
设$d\ge2$且$\mathbf w=(w_1,\ldots,w_d)$满足$w_1\ge\cdots\ge w_d>0$和$\sum_i w_i=1$。令$s_*=d-(1+w_1)^{-1}$。我们证明存在常数$C_{d,\mathbf w}>0$和$\varepsilon_0=\varepsilon_0(d,\mathbf w)>0$，使得对所有$0<\varepsilon<\varepsilon_0$，有$$ \dim_H\operatorname{DI}_{\mathbf w}(\varepsilon)\le s_*+C_{d,\mathbf w}\sqrt\varepsilon。$$结合Kim--Park的下界，这给出了精确公式$$ \dim_H\operatorname{Sing}(\mathbf w)=s_*。$$
英文摘要
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 $$ \dim_H\operatorname{Sing}(\mathbf w)=s_*. $$
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