# Projection theorems for intermediate dimensions

## TL;DR

This paper introduces a new way to understand intermediate dimensions of fractals using capacities, and shows that these dimensions of projections are almost surely independent of the specific subspace chosen.

## Contribution

The paper defines intermediate dimensions via capacities and proves their invariance under projection onto almost all subspaces, linking different fractal dimensions.

## Key findings

- Intermediate dimensions can be characterized using capacities with specific kernels.
- Projections of sets have intermediate dimensions that depend only on the set and the dimension of the subspace.
- A connection between box dimensions of projections and the Hausdorff dimension of the original set is established.

## 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.