Projection theorems for intermediate dimensions

Stuart Andrew Burrell; Kenneth John Falconer; Jonathan MacDonald Fraser

doi:10.4171/JFG/99

Projection theorems for intermediate dimensions

Stuart Andrew Burrell, Kenneth John Falconer, Jonathan MacDonald Fraser

Pure Mathematics

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

Abstract

Intermediate dimensions were recently introduced to interpolate between the Hausdorff and box-counting dimensions of fractals. Firstly, we show that these intermediate dimensions may be defined in terms of capacities with respect to certain kernels. Then, relying on this, we show that the intermediate dimensions of the projection of a set E ⊂ Rⁿ onto almost all m-dimensional subspaces depend only on m and E, that is, they are almost surely independent of the choice of subspace. Our approach is based on ‘intermediate dimension profiles’ which are expressed in terms of capacities. We discuss several applications at the end of the paper, including a surprising result that relates the boxdimensions of the projections of a set to the Hausdorff dimension of the set.

Original language	English
Pages (from-to)	95-116
Number of pages	22
Journal	Journal of Fractal Geometry
Volume	8
Issue number	2
Early online date	30 Apr 2021
DOIs	https://doi.org/10.4171/JFG/99
Publication status	Published - 1 May 2021

Keywords

Intermediate dimensions
Marstrand theorem
Projections
Capacity

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Copyright © 2021, EMS Publishing House. All rights reserved. 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/10.4171/JFG/99.
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Cite this

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title = "Projection theorems for intermediate dimensions",

abstract = "Intermediate dimensions were recently introduced to interpolate between the Hausdorff and box-counting dimensions of fractals. Firstly, we show that these intermediate dimensions may be defined in terms of capacities with respect to certain kernels. Then, relying on this, we show that the intermediate dimensions of the projection of a set E ⊂ Rn onto almost all m-dimensional subspaces depend only on m and E, that is, they are almost surely independent of the choice of subspace. Our approach is based on {\textquoteleft}intermediate dimension profiles{\textquoteright} which are expressed in terms of capacities. We discuss several applications at the end of the paper, including a surprising result that relates the boxdimensions of the projections of a set to the Hausdorff dimension of the set.",

keywords = "Intermediate dimensions, Marstrand theorem, Projections, Capacity",

author = "Burrell, {Stuart Andrew} and Falconer, {Kenneth John} and Fraser, {Jonathan MacDonald}",

note = "Funding: Carnegie Trust (SAB); UK EPSRC Standard Grant (EP/R015104/1) (JMF and KJF).",

year = "2021",

month = may,

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doi = "10.4171/JFG/99",

language = "English",

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T1 - Projection theorems for intermediate dimensions

AU - Burrell, Stuart Andrew

AU - Falconer, Kenneth John

AU - Fraser, Jonathan MacDonald

N1 - Funding: Carnegie Trust (SAB); UK EPSRC Standard Grant (EP/R015104/1) (JMF and KJF).

PY - 2021/5/1

Y1 - 2021/5/1

N2 - Intermediate dimensions were recently introduced to interpolate between the Hausdorff and box-counting dimensions of fractals. Firstly, we show that these intermediate dimensions may be defined in terms of capacities with respect to certain kernels. Then, relying on this, we show that the intermediate dimensions of the projection of a set E ⊂ Rn onto almost all m-dimensional subspaces depend only on m and E, that is, they are almost surely independent of the choice of subspace. Our approach is based on ‘intermediate dimension profiles’ which are expressed in terms of capacities. We discuss several applications at the end of the paper, including a surprising result that relates the boxdimensions of the projections of a set to the Hausdorff dimension of the set.

AB - Intermediate dimensions were recently introduced to interpolate between the Hausdorff and box-counting dimensions of fractals. Firstly, we show that these intermediate dimensions may be defined in terms of capacities with respect to certain kernels. Then, relying on this, we show that the intermediate dimensions of the projection of a set E ⊂ Rn onto almost all m-dimensional subspaces depend only on m and E, that is, they are almost surely independent of the choice of subspace. Our approach is based on ‘intermediate dimension profiles’ which are expressed in terms of capacities. We discuss several applications at the end of the paper, including a surprising result that relates the boxdimensions of the projections of a set to the Hausdorff dimension of the set.

KW - Intermediate dimensions

KW - Marstrand theorem

KW - Projections

KW - Capacity

UR - https://arxiv.org/abs/1907.07632

UR - https://www.ems-ph.org/journals/forthcoming.php?jrn=jfg

UR - http://ems.press/journals/jfg/articles/949030

U2 - 10.4171/JFG/99

DO - 10.4171/JFG/99

M3 - Article

SN - 2308-1309

VL - 8

SP - 95

EP - 116

JO - Journal of Fractal Geometry

JF - Journal of Fractal Geometry

IS - 2

ER -

Projection theorems for intermediate dimensions

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Fourier analytic techniques: Fourier analytic techniques in geometry and analysis

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Projection theorems for intermediate dimensions

Abstract

Keywords

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Fourier analytic techniques: Fourier analytic techniques in geometry and analysis

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