On problems of Danzer and Gowers and dynamics on the space of closed subsets of $\mathbb{R}^d$

Abstract

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