Distance sets, orthogonal projections, and passing to weak tangents
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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.
Fraser , J M 2018 , ' Distance sets, orthogonal projections, and passing to weak tangents ' , Israel Journal of Mathematics , vol. 226 , no. 2 , pp. 851–875 . https://doi.org/10.1007/s11856-018-1715-z
Israel Journal of Mathematics
© 2018, Hebrew University of Jerusalem. This work has been made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1007/s11856-018-1715-z
DescriptionThe author is supported by a Leverhulme Trust Research Fellowship (RF-2016-500).
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