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Attainable values for the Assouad dimensionof projections

Authors: Jonathan M. Fraser and Antti Käenmäki
Journal: Proc. Amer. Math. Soc. 148 (2020), 3393-3405
MSC (2010): Primary 28A80
Published electronically: March 25, 2020
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Abstract: 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 _\mathrm {A} \pi F\forcelinebreak = \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.

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Additional Information

Jonathan M. Fraser
Affiliation: School of Mathematics and Statistics, The University of St Andrews, St Andrews, KY16 9SS, Scotland

Antti Käenmäki
Affiliation: Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, FI-80101 Joensuu, Finland

Keywords: Assouad dimension, orthogonal projection
Received by editor(s): July 9, 2019
Received by editor(s) in revised form: December 15, 2019
Published electronically: March 25, 2020
Additional Notes: The first author was financially supported by a Leverhulme Trust Research Fellowship (RF-2016-500) and an EPSRC Standard Grant (EP/R015104/1).
The second author was financially supported by the Finnish Center of Excellence in Analysis and Dynamics Research and the aforementioned EPSRC grant, which covered the local costs of the visit of the second author to the University of St Andrews in May 2018 where this research began.
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2020 American Mathematical Society