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 $E \subset \mathbb{R}^n$ 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' 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.
Highlights
Theorems on dimensions of projections of fractals in Euclidean space have a long history
In 1954 Marstrand [12] proved that the Hausdorff dimension of the orthogonal projections of a Borel set E ⊂ R2 onto linear subspaces was almost-surely constant
It is natural to seek projection results for the various other dimensions that occur throughout fractal geometry
Summary
Theorems on dimensions of projections of fractals in Euclidean space have a long history. Falconer, Fraser and Kempton [4] introduced intermediate dimensions to provide a continuum of dimensions, one for each θ ∈ [0, 1], that interpolate between the Hausdorff dimension (obtained when θ = 0) and box-counting dimensions (θ = 1). These dimensions are defined by restricting the diameters of sets used in admissible coverings of E to a range [r, rθ] for small r. Some examples and applications are given in the final section
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