Kaufman and Falconer estimates for radial projections and a continuum version of Beck's Theorem

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 on any line, we prove that $$\underset{x\in X}{\sup }{\dim }_{\mathrm{H}}{\pi }_x(Y)\ge \min \{{\dim }_{\mathrm{H}}X,{\dim }_{\mathrm{H}}Y,1\}\mathrm{.}$$ If ${\dim }_{\mathrm{H}}Y>1$, we have the following improved lower bound: $$\underset{x\in X}{\sup }{\dim }_{\mathrm{H}}{\pi }_x(Y\setminus \{x\})\ge \min \{{\dim }_{\mathrm{H}}X+{\dim }_{\mathrm{H}}Y-1,1\}\mathrm{.}$$ 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 ${\dim }_{\mathrm{H}}(X\setminus \mathrm{\ell })={\dim }_{\mathrm{H}}X$ for all lines $\mathrm{\ell }\subset {\mathbb{R}}^2$, then the line set spanned by $X$ has Hausdorff dimension at least $\min \{2{\dim }_{\mathrm{H}}X,2\}$. 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 $\epsilon$-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.