Compact sets with large projections and nowhere dense sumset
Rich\'ard Balka, M\'arton Elekes, Viktor Kiss, Don\'at Nagy, M\'ark, Po\'or

TL;DR
This paper constructs compact sets in Euclidean spaces with large projections but nowhere dense sumsets, answering a question about the relationship between projections and sumset density.
Contribution
It provides a novel random construction of such sets, explores their generalizations, and shows that typical sets in certain spaces share these properties, including a self-similar example.
Findings
Existence of compact sets with full projections and nowhere dense sumsets in higher dimensions.
Most sets with these properties are generic in a Baire category sense.
A self-similar example of such a set in the plane is constructed.
Abstract
We answer a question of Banakh, Jab\l{}o\'nska and Jab\l{}o\'nski by showing that for there exists a compact set such that the projection of onto each hyperplane is of non-empty interior, but is nowhere dense. The proof relies on a random construction. A natural approach in the proofs is to construct such a in the unit cube with full projections, that is, such that the projections of agree with that of the unit cube. We investigate the generalization of these problems for projections onto various dimensional subspaces as well as for -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…
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Taxonomy
TopicsAdvanced Topology and Set Theory · Advanced Banach Space Theory · Limits and Structures in Graph Theory
