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 ‘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.
Citation
Burrell , S A , Falconer , K J & Fraser , J M 2021 , ' Projection theorems for intermediate dimensions ' , Journal of Fractal Geometry , vol. 8 , no. 2 , pp. 95-116 . https://doi.org/10.4171/JFG/99
Publication
Journal of Fractal Geometry
Status
Peer reviewed
DOI
10.4171/JFG/99ISSN
2308-1309Type
Journal article
Description
Funding: Carnegie Trust (SAB); UK EPSRC Standard Grant (EP/R015104/1) (JMF and KJF).Collections
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