A Capacity Approach to Box and Packing Dimensions of Projections and Other Images

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.