Kaufman and Falconer Estimates for Radial Projections and a Continuum Version of Beck’s Theorem

Tuomas Orponen, Pablo Shmerkin, Hong Wang

Research output: Contribution to journalArticlepeer-review


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 dimHY>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 dimH(X ∖ ℓ)=dimHX for all lines, then the line set spanned by X has Hausdorff dimension at least min{2dimHX,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.

Original languageEnglish (US)
Pages (from-to)164-201
Number of pages38
JournalGeometric and Functional Analysis
Issue number1
StatePublished - Feb 2024


  • 28A78
  • 28A80
  • Exceptional sets
  • Hausdorff dimension
  • Radial projections

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology


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