On exceptional sets of radial projections

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 $\dim_{\mathrm{H}} \{x \in \mathbb{R}^{2} \, \setminus \, K : \dim_{\mathrm{H}} π_{x}(K) \leq σ\} \leq \max\{1 + σ- \dim_{\mathrm{H}} K,0\}, \qquad σ\in [0,1).$ If $K \subset \mathbb{R}^{2}$ is a Borel set with $\dim_{\mathrm{H}} K \leq 1$, then $\dim_{\mathrm{H}} \{x \in \mathbb{R}^{2} \, \setminus \, K : \dim_{\mathrm{H}} π_{x}(K) < \dim_{\mathrm{H}} K\} \leq 1.$ 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.

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