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

TL;DR

This paper extends radial projection results in Euclidean spaces using two methods, providing new bounds on the dimensions of projected sets and addressing a conjecture across all dimensions.

Contribution

It introduces a novel approach to radial projections, generalizing previous results and solving a conjecture in all dimensions with two distinct methods.

Findings

Established lower bounds for radial projections in $\

Developed two different proof methods, one shorter and one based on incidence estimates.

Provided new estimates that may be useful for related problems.

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