Projections of totally disconnected thin fractals with very thick shadows on ${\mathbb{R}}^d$

Abstract

We study an extreme scenario of the Marstrand projection theorem for which a fractal has the property that its orthogonal projection is the same as the orthogonal projection of its convex hull. We extend results in current literature and establish checkable criteria for self-affine sets to have this property. Using this, we show that every convex polytope on ${\mathbb{R}}^d$ contains a totally disconnected compact set, which is a union of self-affine sets, of dimension as close to 1 as possible, as well as a rectifiable 1-set, such that the fractal projects to an interval in every 1-dimensional subspace and its convex hull is the given polytope. Other convex sets and projections onto higher dimensional subspaces are also discussed.