Dense forests and Danzer sets

  title={Dense forests and Danzer sets},
  author={Yaar Solomon and Barak Weiss},
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… Expand
On problems of Danzer and Gowers and dynamics on the space of closed subsets of $\mathbb{R}^d$
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 minimalExpand
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Dense forests are discrete subsets of Euclidean space which are uniformly close to all sufficiently long line segments. The degree of density of a dense forest is measured by its visibility function.Expand
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Uniformly Discrete Forests with Poor Visibility
  • N. Alon
  • Computer Science, Mathematics
  • Combinatorics, Probability and Computing
  • 2017
We prove that there is a set F in the plane so that the distance between any two points of F is at least 1, and for any positive ϵ < 1, and every line segment in the plane of length at leastExpand
Arbeitsgemeinschaft mit aktuellem Thema : Mathematical Quasicrystals Mathematisches Forschungsinstitut Oberwolfach October 4-9 2015 Organizers :
Definitions of Delone sets will be given. Standard constructions: cut and project constructions, model sets substitution tilings, return times to a section will be described. For cut and projectExpand
Around the Danzer Problem and the Construction of Dense Forests
A 1965 problem due to Danzer asks whether there exists a set with finite density in Euclidean space intersecting any convex body of volume one. A suitable weakening of the volume constraint leads toExpand


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  • J. Matousek
  • Computer Science, Mathematics
  • Graduate texts in mathematics
  • 2002
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Cet article, issu d'une sorie d'exposes de seminaire faits a FUniversite de Yale au printemps 1967, n'est guere plus qu'une mise au goüt du jour du memoire [5] oü E. Cartan determine lesExpand