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 \[ \text{dim}(\{y\in \mathbb{R}^n \setminus A\mid \text{dim} (π_y(A)) < s\})\leq \max\{k+s -\dim A,0\}. \] The second conjecture is by Liu: Given a Borel set $A\subset {\mathbb{R}}^n$, then \[ \text{dim} (\{x\in \mathbb{R}^n \setminus A \mid \text{dim}(π_x(A))<\text{dim} A\}) \leq \lceil \text{dim} A\rceil. \]

31 pages; we deleted two sections in the previous version (about the new proofs on classical results); we added more details