Hausdorff Dimension of Weighted Singular Vectors

Abstract

Let $d\ge 2$ and let $\mathbf{w}=(w_1,\dots ,w_d)$ satisfy $w_1\ge \cdots \ge w_d>0$ and ${\sum }_i{w}_i=1$. Set $s_{\ast }=d-(1+w_1{)}^{-1}$. We prove that there exist constants $C_{d,\mathbf{w}}>0$ and ${\epsilon }_0={\epsilon }_0(d,\mathbf{w})>0$ such that for all $0<\epsilon <{\epsilon }_0$, $${\dim }_H{\mathrm{DI}}_{\mathbf{w}}(\epsilon )\le s_{\ast }+C_{d,\mathbf{w}}\sqrt{\epsilon }.$$ Together with the lower bound of Kim--Park, this gives the exact formula $${\dim }_H\mathrm{Sing}(\mathbf{w})=s_{\ast }.$$

arXiv comment18 pages