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

TL;DR

This paper investigates the dynamics of closed subsets of  under affine transformations, resolving a Gowers question about Danzer sets and establishing bounds on  point distributions within convex sets.

Contribution

It characterizes minimal subsystems in the space of closed sets, proving the non-existence of certain Danzer sets and providing quantitative bounds on point distributions in convex sets.

Findings

Only trivial minimal subsystems are fixed points  and empty set.

No Danzer set exists with bounded intersections across all convex sets of volume one.

Any -net for convex sets contains a small volume convex set with many net points.

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 $N$.