Exceptional set estimates for radial projections in $\mathbb{R}^n$

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})\leq \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}) \leq \lceil \text{dim} A\rceil. ]