Compact sets with large projections and nowhere dense sumset

Abstract

We answer a question of Banakh, Jab\l{}o\'nska and Jab\l{}o\'nski by showing that for $d\ge 2$ there exists a compact set $K \subseteq \mathbb{R}^d$ such that the projection of $K$ onto each hyperplane is of non-empty interior, but $K+K$ is nowhere dense. The proof relies on a random construction. A natural approach in the proofs is to construct such a $K$ in the unit cube with full projections, that is, such that the projections of $K$ agree with that of the unit cube. We investigate the generalization of these problems for projections onto various dimensional subspaces as well as for $\ell$-fold sumsets. We obtain numerous positive and negative results, but also leave open many interesting cases. We also show that in most cases if we have a specific example of such a compact set then actually the generic (in the sense of Baire category) compact set in a suitably chosen space is also an example. Finally, utilizing a computer-aided construction, we show that the compact set in the plane with full projections and nowhere dense sumset can be self-similar.