Dense forests and Danzer sets

@article{Solomon2014DenseFA,
  title={Dense forests and Danzer sets},
  author={Yaar Solomon and Barak Weiss},
  journal={ArXiv},
  year={2014},
  volume={abs/1406.3807}
}
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… 
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