# A capacity approach to box and packing dimensions of projections of sets and exceptional directions

## TL;DR

This paper introduces a capacity-based approach to analyze the box and packing dimensions of projections of sets in Euclidean space, providing clearer formulas, estimates on exceptional sets, and a unified framework for different dimensions.

## Contribution

It offers new capacity-based definitions of dimension profiles, simplifies the computation of projection dimensions, and unifies various projection results using Fourier analysis.

## Key findings

- Capacity-based formulas for projection dimensions
- Estimates on size of exceptional subspaces
- Unified approach for different dimension types

## Abstract

Dimension profiles were introduced in [8,11] to give a formula for the box-counting and packing dimensions of the orthogonal projections of a set $R^n$ onto almost all $m$-dimensional subspaces. However, these definitions of dimension profiles are indirect and are hard to work with. Here we firstly give alternative definitions of dimension profiles in terms of capacities of $E$ with respect to certain kernels, which lead to the box-counting and packing dimensions of projections fairly easily, including estimates on the size of the exceptional sets of subspaces where the dimension of projection is smaller the typical value. Secondly, we argue that with this approach projection results for different types of dimension may be thought of in a unified way. Thirdly, we use a Fourier transform method to obtain further inequalities on the size of the exceptional subspaces.