Distance sets, orthogonal projections, and passing to weak tangents

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Abstract

We consider the Assouad dimension analogues of two important problems in geometric measure theory. These problems are tied together by the common theme of ‘passing to weak tangents’. First, we solve the analogue of Falconer’s distance set problem for Assouad dimension in the plane: if a planar set has Assouad dimension greater than 1, then its distance set has Assouad dimension 1. We also obtain partial results in higher dimensions. Second, we consider how Assouad dimension behaves under orthogonal projection. We extend the planar projection theorem of Fraser and Orponen to higher dimensions, provide estimates on the (Hausdorff) dimension of the exceptional set of projections, and provide a recipe for obtaining results about restricted families of projections. We provide several illustrative examples throughout.
Original languageEnglish
Pages (from-to)851–875
Number of pages25
JournalIsrael Journal of Mathematics
Volume226
Issue number2
Early online date8 Jun 2018
DOIs
Publication statusPublished - Jun 2018

Keywords

  • Assouad dimension
  • Weak tangent
  • Distance set
  • Orthogonal projection
  • Exceptional set
  • Restricted families

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