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 be non-empty Borel sets. If X is not contained in any line, we prove that (Formula presented.) If dim_HY>1, we have the following improved lower bound: (Formula presented.) 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 is a Borel set with the property that dim_H(X ∖ ℓ)=dim_HX for all lines, then the line set spanned by X has Hausdorff dimension at least min{2dim_HX,2}. While the results above concern, we also derive some counterparts in 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.