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

Abstract

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 $X\ne \varnothing$. If $\dim Y\in (k,k+1]$ for some $k\in \{1,\dots ,n-1\}$, then \[ \sup_{x\in X} \dim π_x(Y\setminus \{x\}) \geq \min \{\dim X + \dim Y - k, k\}. \] 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.

17 pages