On exceptional sets of radial projections

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 $${\dim }_{\mathrm{H}}\{x\in {\mathbb{R}}^2\setminus K:{\dim }_{\mathrm{H}}{\pi }_x(K)\le \sigma \}\le \max \{1+\sigma -{\dim }_{\mathrm{H}}K,0\},\sigma \in [0,1).$$ If $K\subset {\mathbb{R}}^2$ is a Borel set with ${\dim }_{\mathrm{H}}K\le 1$, then $${\dim }_{\mathrm{H}}\{x\in {\mathbb{R}}^2\setminus K:{\dim }_{\mathrm{H}}{\pi }_x(K)<{\dim }_{\mathrm{H}}K\}\le 1.$$ 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.

25 pages