A capacity approach to box and packing dimensions of projections of sets and exceptional directions
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.
Citation
Falconer , K J 2021 , ' A capacity approach to box and packing dimensions of projections of sets and exceptional directions ' , Journal of Fractal Geometry , vol. 8 , no. 1 , pp. 1-26 . https://doi.org/10.4171/JFG/96
Publication
Journal of Fractal Geometry
Status
Peer reviewed
ISSN
2308-1309Type
Journal article
Rights
© 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 © 2021 European Mathematical Society Published by EMS Press This work is licensed under a CC BY 4.0 license
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