A nonlinear projection theorem for Assouad dimension and applications

Jonathan Fraser*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)
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We prove a general nonlinear projection theorem for Assouad dimension. This theorem has several applications including to distance sets, radial projections, and sum-product phenomena. In the setting of distance sets we are able to completely resolve the planar version of Falconer’s distance set problem for Assouad dimension, both dealing with the awkward ‘critical case’ and providing sharp estimates for sets with Assouad dimension less than 1. In the higher dimensional setting we connect the problem to the dimension of the set of exceptions in a related (orthogonal) projection theorem. We also obtain results on pinned distance sets and our results still hold when distances are taken with respect to a sufficiently curved norm. As another application we prove a radial projection theorem for Assouad dimension with sharp estimates on the Hausdorff dimension of the exceptional set.
Original languageEnglish
Pages (from-to)777-797
Number of pages21
JournalJournal of the London Mathematical Society
Issue number2
Early online date8 Dec 2022
Publication statusPublished - 1 Feb 2023


  • Assouad dimension
  • Nonlinear projections
  • Distance sets
  • Radial projections
  • Exceptional set
  • Hausdorff dimension
  • Sum-product theorem


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