On Hausdorff dimension of radial projections

For any $x\in {\mathbb{R}}^d$, $d\ge 2$, denote $\pi^x: $\mathbb{R}$^d\backslash{x}\rightarrow S^{d-1}$astheradialprojection$${\pi }^x(y)=\frac{y-x}{\mathrm{\mid }y-x\mathrm{\mid }}\mathrm{.}$$GivenaBorelset$E\subset{\Bbb R}^d$,$\dim_{$\mathcal{H}$} E\leq d-1$, in this paper we investigate for how many $x\in {\mathbb{R}}^d$ the radial projection ${\pi }^x$ preserves the Hausdorff dimension of $E$, namely whether ${\dim }_{\mathcal{H}}{\pi }^x(E)={\dim }_{\mathcal{H}}E$. We develop a general framework to link ${\pi }^x(E)$, $x\in F$ and ${\pi }^y(F)$, $y\in E$, for any Borel set $F\subset {\mathbb{R}}^d$. In particular, whether ${\dim }_{\mathcal{H}}{\pi }^x(E)={\dim }_{\mathcal{H}}E$ for some $x\in F$ can be reduced to whether $F$ is visible from some $y\in E$ (i.e. ${\mathcal{H}}^{d-1}({\pi }^y(F))>0$). This allows us to apply Orponen’s estimate on visibility to obtain $${\dim }_{\mathcal{H}}\{x\in {\mathbb{R}}^d:{\dim }_{\mathcal{H}}{\pi }^x(E)<{\dim }_{\mathcal{H}}E\}\le 2(d-1)-{\dim }_{\mathcal{H}}E,$$ for any Borel set $E\subset {\mathbb{R}}^d$, ${\dim }_{\mathcal{H}}E\in (d-2,d-1]$. This improves the Peres-Schlag bound when ${\dim }_{\mathcal{H}}E\in (d-\frac{3}{2},d-1]$, and it is optimal at the endpoint ${\dim }_{\mathcal{H}}E=d-1$.