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

Abstract

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 \[ \sup_{x \in X} \dim_{\mathrm{H}} π_{x}(Y) \geq \min\{\dim_{\mathrm{H}} X,\dim_{\mathrm{H}} Y,1\}. \] If ${\dim }_{\mathrm{H}}Y>1$, we have the following improved lower bound: \[ \sup_{x \in X} \dim_{\mathrm{H}} π_{x}(Y \, \setminus \, \{x\}) \geq \min\{\dim_{\mathrm{H}} X + \dim_{\mathrm{H}} Y - 1,1\}. \] 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 ${\dim }_{\mathrm{H}}(X\setminus \ell )={\dim }_{\mathrm{H}}X$ for all lines $\ell \subset {\mathbb{R}}^2$, 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.

31 pages. This paper supersedes arXiv:2205.13890

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