Projection theorems for intermediate dimensions

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

https://doi.org/10.4171/JFG/99

ISSN

2308-1309

Type

Journal article

Rights

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.

Copyright © 2021 European Mathematical Society. This is an open access article under the terms of the Creative Commons Attribution 4.0 License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Description

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