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

TL;DR

This paper proves two conjectures related to the dimensions of radial projections of Borel sets in Euclidean space, establishing bounds on the size of exceptional sets where projections have smaller than expected dimension.

Contribution

The paper confirms two conjectures on radial projection dimensions, advancing understanding of geometric measure theory and projection properties of fractal sets.

Findings

Proved Lund, Pham, and Thu's conjecture on dimension bounds for radial projections.

Confirmed Liu's conjecture on the dimension of exceptional sets for radial projections.

Established new bounds on the size of sets where radial projections have reduced dimension.

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