The box dimensions of exceptional self-affine sets in ℝ3

Jonathan Fraser; Natalia Anna Jurga

doi:10.1016/j.aim.2021.107734

The box dimensions of exceptional self-affine sets in ℝ3

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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.

Original language	English
Article number	107734
Number of pages	32
Journal	Advances in Mathematics
Volume	385
Early online date	28 Apr 2021
DOIs	https://doi.org/10.1016/j.aim.2021.107734
Publication status	Published - 16 Jul 2021

Keywords

Self-affine set
Box dimension
Singular values

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10.1016/j.aim.2021.107734

Fraser-2021-Box-dimension-of-AdvMath-AAM
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.
Accepted author manuscript, 418 KB

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis
Fraser, J. (PI) & Falconer, K. J. (CoI)
EPSRC
1/02/18 → 11/06/21
Project: Standard

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title = "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.",

keywords = "Self-affine set, Box dimension, Singular values",

author = "Jonathan Fraser and Jurga, {Natalia Anna}",

note = "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).",

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AB - 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.

KW - Self-affine set

KW - Box dimension

KW - Singular values

U2 - 10.1016/j.aim.2021.107734

DO - 10.1016/j.aim.2021.107734

M3 - Article

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VL - 385

JO - Advances in Mathematics

JF - Advances in Mathematics

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ER -

The box dimensions of exceptional self-affine sets in ℝ3

Abstract

Keywords

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Fourier analytic techniques: Fourier analytic techniques in geometry and analysis

Cite this