Exceptional set estimates for radial projections in R^n

Abstract

We prove two conjectures in this paper. The first conjecture is by Lund, Pham and Thu: Given a Borel set $A\subset {\mathbb{R}}^n$ such that $\dim A\in (k,k+1]$ for some $k\in \{1,\dots ,n-1\}$. For $0<s<k$, we have   $\text{dim}(\{y\in {\mathbb{R}}^n\setminus A\mid \text{dim}({\pi }_y(A))<s\})\le max\{k+s-\dim A,0\}.$   The second conjecture is by Liu: Given a Borel set $A\subset {\mathbb{R}}^n$, then
$\text{dim}(\{x\in {\mathbb{R}}^n\setminus A\mid \text{dim}({\pi }_x(A))<\text{dim}A\})\le \lceil \text{dim}A\rceil .$