  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.
