On exceptional sets of radial projections

TL;DR

This paper establishes new bounds on the size of sets of points in the plane from which the radial projections of a given set have small Hausdorff dimension, resolving conjectures in the field.

Contribution

It provides the first planar cases of conjectures by Lund-Thang-Huong Thu and Liu regarding exceptional radial projections for sets with various Hausdorff dimensions.

Findings

For sets with Hausdorff dimension greater than 1, bounds on the dimension of points with small projection dimension.

For sets with Hausdorff dimension at most 1, the dimension of points with projections of smaller dimension is at most 1.

The results match finite field analogues and resolve key conjectures in geometric measure theory.

Abstract

We prove two new exceptional set estimates for radial projections in the plane. If $K \subset \mathbb{R}^{2}$ is a Borel set with $\dim_{\mathrm{H}} K > 1$, then $$\dim_{\mathrm{H}} \{x \in \mathbb{R}^{2} \, \setminus \, K : \dim_{\mathrm{H}} \pi_{x}(K) \leq \sigma\} \leq \max\{1 + \sigma - \dim_{\mathrm{H}} K,0\}, \qquad \sigma \in [0,1).$$ If $K \subset \mathbb{R}^{2}$ is a Borel set with $\dim_{\mathrm{H}} K \leq 1$, then $$\dim_{\mathrm{H}} \{x \in \mathbb{R}^{2} \, \setminus \, K : \dim_{\mathrm{H}} \pi_{x}(K) < \dim_{\mathrm{H}} K\} \leq 1.$$ The finite field counterparts of both results above were recently proven by Lund, Thang, and Huong Thu. Our results resolve the planar cases of conjectures of Lund-Thang-Huong Thu, and Liu.