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

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 $E\subset {\mathbb{R}}^n$ 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 boxcounting 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 than 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.