Assouad dimension influences the box and packing dimensions of orthogonal projections

Kenneth John Falconer; Jonathan Fraser; Pablo Shmerkin

doi:10.4171/jfg/104

Assouad dimension influences the box and packing dimensions of orthogonal projections

Kenneth John Falconer, Jonathan Fraser, Pablo Shmerkin

Pure Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We present several applications of the Assouad dimension, and the related quasi-Assouad dimension and Assouad spectrum, to the box and packing dimensions of orthogonal projections of sets. For example, we show that if the (quasi-)Assouad dimension of F ⊆ Rⁿ is no greater than m, then the box and packing dimensions of F are preserved under orthogonal projections onto almost all m-dimensional subspaces. We also show that the threshold m for the (quasi-)Assouad dimension is sharp, and bound the dimension of the exceptional set of projections strictly away from the dimension of the Grassmannian.

Original language	English
Pages (from-to)	247–259
Journal	Journal of Fractal Geometry
Volume	8
Issue number	3
Early online date	3 May 2021
DOIs	https://doi.org/10.4171/jfg/104
Publication status	Published - 2021

Keywords

Assouad dimension
Box dimension
Packing dimension
Orthogonal projection

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10.4171/jfg/104Licence: CC BY

Falconer-2021-Assouad-dimension-influences-JFG-8-3-247-CCBY
Copyright © 2021 European Mathematical Society Published by EMS Press This work is licensed under a CC BY 4.0 license.
Final published version, 165 KBLicence: CC BY

New perspectives in the dimension: New perspectives in the dimension theory of fractals
Fraser, J. (PI)
The Leverhulme Trust
1/09/19 → 31/01/23
Project: Standard
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

Cite this

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title = "Assouad dimension influences the box and packing dimensions of orthogonal projections",

abstract = "We present several applications of the Assouad dimension, and the related quasi-Assouad dimension and Assouad spectrum, to the box and packing dimensions of orthogonal projections of sets. For example, we show that if the (quasi-)Assouad dimension of F ⊆ Rn is no greater than m, then the box and packing dimensions of F are preserved under orthogonal projections onto almost all m-dimensional subspaces. We also show that the threshold m for the (quasi-)Assouad dimension is sharp, and bound the dimension of the exceptional set of projections strictly away from the dimension of the Grassmannian.",

keywords = "Assouad dimension, Box dimension, Packing dimension, Orthogonal projection",

author = "Falconer, {Kenneth John} and Jonathan Fraser and Pablo Shmerkin",

note = "Funding: UK EPSRC Standard Grant (EP/R015104/1) (KJF and JMF). Leverhulme Trust Research Project Grant (RPG-2019-034) (JMF). Royal Society International Exchange grant IES\R1\191195 (KJF and PS). ProjectPICT 2015-3675 (ANPCyT) (PS).",

year = "2021",

doi = "10.4171/jfg/104",

language = "English",

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journal = "Journal of Fractal Geometry",

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T1 - Assouad dimension influences the box and packing dimensions of orthogonal projections

AU - Falconer, Kenneth John

AU - Fraser, Jonathan

AU - Shmerkin, Pablo

N1 - Funding: UK EPSRC Standard Grant (EP/R015104/1) (KJF and JMF). Leverhulme Trust Research Project Grant (RPG-2019-034) (JMF). Royal Society International Exchange grant IES\R1\191195 (KJF and PS). ProjectPICT 2015-3675 (ANPCyT) (PS).

PY - 2021

Y1 - 2021

N2 - We present several applications of the Assouad dimension, and the related quasi-Assouad dimension and Assouad spectrum, to the box and packing dimensions of orthogonal projections of sets. For example, we show that if the (quasi-)Assouad dimension of F ⊆ Rn is no greater than m, then the box and packing dimensions of F are preserved under orthogonal projections onto almost all m-dimensional subspaces. We also show that the threshold m for the (quasi-)Assouad dimension is sharp, and bound the dimension of the exceptional set of projections strictly away from the dimension of the Grassmannian.

AB - We present several applications of the Assouad dimension, and the related quasi-Assouad dimension and Assouad spectrum, to the box and packing dimensions of orthogonal projections of sets. For example, we show that if the (quasi-)Assouad dimension of F ⊆ Rn is no greater than m, then the box and packing dimensions of F are preserved under orthogonal projections onto almost all m-dimensional subspaces. We also show that the threshold m for the (quasi-)Assouad dimension is sharp, and bound the dimension of the exceptional set of projections strictly away from the dimension of the Grassmannian.

KW - Assouad dimension

KW - Box dimension

KW - Packing dimension

KW - Orthogonal projection

U2 - 10.4171/jfg/104

DO - 10.4171/jfg/104

M3 - Article

SN - 2308-1309

VL - 8

SP - 247

EP - 259

JO - Journal of Fractal Geometry

JF - Journal of Fractal Geometry

IS - 3

ER -

Assouad dimension influences the box and packing dimensions of orthogonal projections

Abstract

Keywords

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Projects

New perspectives in the dimension: New perspectives in the dimension theory of fractals

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis

Cite this