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

Kenneth John Falconer*

*Corresponding author for this work

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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 languageEnglish
Pages (from-to)1-26
Number of pages26
JournalJournal of Fractal Geometry
Volume8
Issue number1
Early online date7 Dec 2020
DOIs
Publication statusPublished - 8 Mar 2021

Keywords

  • Projection
  • Box dimension
  • Packing dimension
  • Hausdorff dimension
  • Capacity

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