Radial Projections in ${\mathbb{R}}^n$ Revisited

We generalize the recent results on radial projections by Orponen, Shmerkin, Wang using two different methods. In particular, we show that given $X,Y\subset {\mathbb{R}}^n$ Borel sets and $X\mathrm{\ne }\mathrm{\varnothing }$. If $\dim Y\in (k,k+1]$ for some $k\in \{1,\dots ,n-1\}$, then $$\underset{x\in X}{\sup }\dim {\pi }_x(Y\setminus \{x\})\ge \min \{\dim X+\dim Y-k,k\}\mathrm{.}$$ Our results give a new approach to solving a conjecture of Lund-Pham-Thu in all dimensions and for all ranges of $\dim Y$. The first of our two methods for proving the above theorem is shorter, utilizing a result of the first author and Gan. Our second method, though longer, follows the original methodology of Orponen–Shmerkin–Wang, and requires a higher dimensional incidence estimate and a dual Furstenberg-set estimate for lines. These new estimates may be of independent interest.