We provide several new answers on the question: how do radial projections distort the dimension of planar sets? Let $X,Y \subset \mathbb{R}^{2}$ be non-empty Borel sets. If X is not contained in any line, we prove that \( \sup _{x \in X} \dim _{\mathrm {H}}\pi _{x}(Y \, \setminus \, \{x\}) \geq \min \{ \dim _{\mathrm {H}}X,\dim _{\mathrm {H}}Y,1\}. \) If dimHY>1, we have the following improved lower bound: \( \sup _{x \in X} \dim _{\mathrm {H}}\pi _{x}(Y \, \setminus \, \{x\}) \geq \min \{ \dim _{\mathrm {H}}X + \dim _{\mathrm {H}}Y - 1,1\}. \) Our results solve conjectures of Lund-Thang-Huong, Liu, and the first author. Another corollary is the following continuum version of Beck’s theorem in combinatorial geometry: if $X \subset \mathbb{R}^{2}$ is a Borel set with the property that dimH(X ∖ ℓ)=dimHX for all lines $\ell \subset \mathbb{R}^{2}$ , then the line set spanned by X has Hausdorff dimension at least min{2dimHX,2}.
While the results above concern $\mathbb{R}^{2}$ , we also derive some counterparts in $\mathbb{R}^{d}$ by means of integralgeometric considerations. The proofs are based on an ϵ-improvement in the Furstenberg set problem, due to the two first authors, a bootstrapping scheme introduced by the second and third author, and a new planar incidence estimate due to Fu and Ren.
