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

Figures (4)

• Figure 1: $\mathbb T_{y,j}$ in the radial projection
• Figure 2: Dual Slabs
• Figure 3: $\mathbb T_{y,\Delta}$ in the radial projection
• Figure 4: $\mathbb T_{x,j}$ in the radial projection

Theorems & Definitions (39)

• Theorem 1
• Theorem 2
• Definition 3
• Definition 4
• Remark 5
• Remark 6
• Lemma 7
• proof
• Lemma 8
• proof