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 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 dim_HY>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 dim_H(X ∖ ℓ)=dim_HX for all lines \(\ell \subset \mathbb{R}^{2}\), then the line set spanned by X has Hausdorff dimension at least min{2dim_HX,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.