On Problems of Danzer and Gowers and Dynamics on the Space of Closed Subsets of R

Considering the space of closed subsets of R, endowed with the Chabauty-Fell topology, and the affine action of SLd(R)nR, we prove that the only minimal subsystems are the fixed points {∅} and {R}. As a consequence we resolve a question of Gowers concerning the existence of certain Danzer sets: there is no set Y ⊂ R such that for every convex set C ⊂ R 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] there is a convex set of volume ε containing at least Ω(log log(1/ε)) points of N .