A capacity approach to box and packing dimensions of projections of sets and exceptional directions

Kenneth John Falconer

doi:10.4171/JFG/96

A capacity approach to box and packing dimensions of projections of sets and exceptional directions

Kenneth John Falconer^*

^*Corresponding author for this work

Pure Mathematics

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

Abstract

Dimension profiles were introduced in [8,11] to give a formula for the box-counting and packing dimensions of the orthogonal projections of a set E in ℝⁿ onto almost all m-dimensional subspaces. However, these definitions of dimension profiles are indirect and are hard to work with. Here we firstly give alternative definitions of dimension profiles in terms of capacities of E with respect to certain kernels, which lead to the box-counting and packing dimensions of projections fairly easily, including estimates on the size of the exceptional sets of subspaces where the dimension of projection is smaller the typical value. Secondly, we argue that with this approach projection results for different types of dimension may be thought of in a unified way. Thirdly, we use a Fourier transform method to obtain further inequalities on the size of the exceptional subspaces.

Original language	English
Pages (from-to)	1-26
Number of pages	26
Journal	Journal of Fractal Geometry
Volume	8
Issue number	1
Early online date	7 Dec 2020
DOIs	https://doi.org/10.4171/JFG/96
Publication status	Published - 8 Mar 2021

Keywords

Projection
Box dimension
Packing dimension
Hausdorff dimension
Capacity

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10.4171/JFG/96

CapacityProjections-Falconer
© 2019, European Mathematical Society. This work has been made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at www.ems-ph.org
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Cite this

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AB - Dimension profiles were introduced in [8,11] to give a formula for the box-counting and packing dimensions of the orthogonal projections of a set E in ℝn onto almost all m-dimensional subspaces. However, these definitions of dimension profiles are indirect and are hard to work with. Here we firstly give alternative definitions of dimension profiles in terms of capacities of E with respect to certain kernels, which lead to the box-counting and packing dimensions of projections fairly easily, including estimates on the size of the exceptional sets of subspaces where the dimension of projection is smaller the typical value. Secondly, we argue that with this approach projection results for different types of dimension may be thought of in a unified way. Thirdly, we use a Fourier transform method to obtain further inequalities on the size of the exceptional subspaces.

KW - Projection

KW - Box dimension

KW - Packing dimension

KW - Hausdorff dimension

KW - Capacity

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A capacity approach to box and packing dimensions of projections of sets and exceptional directions

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