Projection theorems for intermediate dimensions
Stuart A. Burrell
University of St Andrews, UKKenneth J. Falconer
University of St Andrews, UKJonathan M. Fraser
University of St Andrews, UK
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 onto almost all -dimensional subspaces depend only on and , that is, they are almost surely independent of the choice of subspace. Our approach is based on that are expressed in terms of capacities. We discuss several applications at the end of the paper, including a surprising result that relates the box dimensions of the projections of a set to the Hausdorff dimension of the set.
Cite this article
Stuart A. Burrell, Kenneth J. Falconer, Jonathan M. Fraser, Projection theorems for intermediate dimensions. J. Fractal Geom. 8 (2021), no. 2, pp. 95–116
DOI 10.4171/JFG/99