A capacity approach to box and packing dimensions of projections and other images

TL;DR

This paper introduces a capacity-based approach to compute the box-counting and packing dimensions of projections and images of sets, simplifying the original dimension profile theory and extending its applications.

Contribution

It redefines dimension profiles using capacities, making the calculation of dimensions of projections and images more straightforward and connecting it to packing dimension theory.

Findings

Capacity-based dimension profiles simplify calculations.

The approach provides insights into exceptional sets of projections.

Connections to packing dimension are established.

Abstract

Dimension profiles were introduced by Falconer and Howroyd to provide formulae for the box-counting and packing dimensions of the orthogonal projections of a set E or a measure on Euclidean space onto almost all m-dimensional subspaces. The original definitions of dimension profiles are somewhat awkward and not easy to work with. Here we rework this theory with an alternative definition of dimension profiles in terms of capacities of E with respect to certain kernels, and this leads to the box-counting dimensions of projections and other images of sets relatively easily. We also discuss other uses of the profiles, such as the information they give on exceptional sets of projections and dimensions of images under certain stochastic processes. We end by relating this approach to packing dimension.