Exceptional set estimates for radial projections in R^n

Abstrakti

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
\(\text{dim} (\{x\in \mathbb{R}^n \setminus A \mid \text{dim}(\pi_x(A))<\text{dim} A\}) \leq \lceil \text{dim} A\rceil.\)