Attainable values for the Assouad dimension of projections

We prove that for an arbitrary upper semi-continuous function \(\phi\colon G(1,2) \to [0,1]\) there exists a compact set \(F\) in the plane such that \(\dim_{\textrm{A}} \pi F = \phi(\pi)\) for all \(\pi \in G(1,2)\), where \(\pi F\) is the orthogonal projection of \(F\) onto the line \(\pi\). In pa...

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Bibliographic Details
Published in:arXiv.org 2020-01
Main Authors: Fraser, Jonathan M, Käenmäki, Antti
Format: Article
Language:English
Subjects:
Colon
Continuity (mathematics)
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Summary:We prove that for an arbitrary upper semi-continuous function \(\phi\colon G(1,2) \to [0,1]\) there exists a compact set \(F\) in the plane such that \(\dim_{\textrm{A}} \pi F = \phi(\pi)\) for all \(\pi \in G(1,2)\), where \(\pi F\) is the orthogonal projection of \(F\) onto the line \(\pi\). In particular, this shows that the Assouad dimension of orthogonal projections can take on any finite or countable number of distinct values on a set of projections with positive measure. It was previously known that two distinct values could be achieved with positive measure. Recall that for other standard notions of dimension, such as the Hausdorff, packing, upper or lower box dimension, a single value occurs almost surely.
ISSN:2331-8422