On problems of Danzer and Gowers and dynamics on the space of closed subsets of Rᵈ

Original · EN

Considering the space of closed subsets of Rᵈ, endowed with the Chabauty-Fell topology, and the affine action of SLd(R)ᵈ, we prove that the only minimal subsystems are the fixed points {} and {Rᵈ}. As a consequence we resolve a question of Gowers concerning the existence of certain Danzer sets: there is no set Y ⊂ Rᵈ such that for every convex set C ⊂ Rᵈ of volume one, the cardinality of C ∩ Y is bounded above and below by nonzero contants independent of C. We also provide a short independent proof of this fact and deduce a quantitative consequence: for every ε-net N for convex sets in [0,1]ᵈ there is a convex set of volume ε containing at least Ω((1/ε)) points of N.