Attainable values for the Assouad dimension of projections
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- by Jonathan M. Fraser and Antti Käenmäki PDF
- Proc. Amer. Math. Soc. 148 (2020), 3393-3405 Request permission
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 $\dim _\mathrm {A} \pi F = \phi (\pi )$ 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.References
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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
- MR Author ID: 946983
- Email: jmf32@st-andrews.ac.uk
- Antti Käenmäki
- Affiliation: Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, FI-80101 Joensuu, Finland
- Email: antti.kaenmaki@uef.fi
- 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
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 3393-3405
- MSC (2010): Primary 28A80
- DOI: https://doi.org/10.1090/proc/14999
- MathSciNet review: 4108846