Distance sets, orthogonal projections and passing to weak tangents

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 h...

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Bibliographic Details
Published in:Israel journal of mathematics 2018-06, Vol.226 (2), p.851-875
Main Author: Fraser, Jonathan M.
Format: Article
Language:English
Subjects:
Algebra
Analysis
Applications of Mathematics
Group Theory and Generalizations
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Tangents
Theoretical
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Summary: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.
ISSN:0021-2172
1565-8511
DOI:10.1007/s11856-018-1715-z