Dense forests and Danzer sets

TL;DR

This paper investigates the existence and properties of Danzer sets in Euclidean space, proving limitations of natural constructions, exploring dense forests, and establishing bounds on their growth rates using advanced dynamics and combinatorics.

Contribution

It demonstrates that common candidate sets are not Danzer sets, constructs dense forests via homogeneous dynamics, and improves bounds on Danzer set growth rates.

Findings

Natural candidates are not Danzer sets

Existence of dense forests with specific properties

Improved growth rate bounds for Danzer sets

Abstract

A set $Y\subseteq\mathbb{R}^d$ that intersects every convex set of volume $1$ is called a Danzer set. It is not known whether there are Danzer sets in $\mathbb{R}^d$ with growth rate $O(T^d)$. We prove that natural candidates, such as discrete sets that arise from substitutions and from cut-and-project constructions, are not Danzer sets. For cut and project sets our proof relies on the dynamics of homogeneous flows. We consider a weakening of the Danzer problem, the existence of uniformly discrete dense forests, and we use homogeneous dynamics (in particular Ratner's theorems on unipotent flows) to construct such sets. We also prove an equivalence between the above problem and a well-known combinatorial problem, and deduce the existence of Danzer sets with growth rate $O(T^d\log T)$, improving the previous bound of $O(T^d\log^{d-1} T)$.