Compact sets with large projections and nowhere dense sumset

Balka, Richárd and Elekes, Márton and Kiss, Viktor and Nagy, Donát and Poór, Márk (2023) Compact sets with large projections and nowhere dense sumset. NONLINEARITY, 36 (10). pp. 5190-5215. ISSN 0951-7715

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Abstract

We answer a question of Banakh, Jab lo´nska and Jab lo´nski by showing that for d ≥ 2 there exists a compact set K ⊆ Rd 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 ℓ-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 selfsimilar.

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