The box dimensions of exceptional self-affine sets in ℝ3
Abstract
We study the box dimensions of self-affine sets in ℝ3 which are generated by afinite collection of generalised permutation matrices. We obtain bounds for the dimensions which hold with very minimal assumptions and give rise to sharp results in many cases. There are many issues in extending the well-established planar theory to R3 including that the principal planar projections are (affine distortions of) self-affine sets with overlaps (rather than self-similar sets) and that the natural modified singular value function fails to be sub-multiplicative in general. We introduce several new techniques to deal with these issues and hopefully provide some insight into the challenges in extending the theory further.
Citation
Fraser , J & Jurga , N A 2021 , ' The box dimensions of exceptional self-affine sets in ℝ3 ' , Advances in Mathematics , vol. 385 , 107734 . https://doi.org/10.1016/j.aim.2021.107734
Publication
Advances in Mathematics
Status
Peer reviewed
ISSN
0001-8708Type
Journal article
Rights
Crown Copyright © 2021 Published by Elsevier. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted 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/10.1016/j.aim.2021.107734.
Description
Funding: JMF was financially supported by an EPSRC Standard Grant (EP/R015104/1). NJ was financially supported by a Leverhulme Trust Research Project Grant (RPG-2016-194).Collections
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