2024

Orponen, Tuomas; Shmerkin, Pablo; Wang, Hong

We provide several new answers on the question: how do radial projections distort the dimension of planar sets? Let X,Y ⊂ R2 be non-empty Borel sets. If X is not contained in any line, we prove that sup x∈X dimH πx(Y \ {x}) ≥ min{dimH X, dimH Y, 1}. If dimH Y > 1, we have the following improved lower bound: sup x∈X dimH πx(Y \ {x}) ≥ min{dimH X + dimH 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 ⊂ R2 is a Borel set with the property that dimH(X \ ) = dimH X for all lines ⊂ R2, then the line set spanned by X has Hausdorff dimension at least min{2dimH X, 2}. While the results above concern R2, we also derive some counterparts in Rd 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.