论文标题
在丹泽问题和茂密森林的建造周围
Around the Danzer Problem and the Construction of Dense Forests
论文作者
论文摘要
1965年由于丹泽（Danzer）引起的问题询问是否存在欧几里得空间中有限密度的集合，与第一卷的任何凸体相交。合适的体积约束削弱会导致（最近）构建\ emph {茂密的森林}的问题。这些是离散的点集，均匀地接近足够长的线段。 到目前为止，朝着这些问题的进展涉及围绕组合和计算几何形状，凸几何形状，寄养者近似，差异理论，动力学系统理论，指数总和，傅立叶分析，同质动力学，Quasicicrystals和Pliposicy理论的数学理论。 本文的目的是调查与Danzer问题和建造茂密森林有关的已知结果，概括其中的一些结果，并提出许多开放问题，以进一步进步，以解决这个长期以来的问题。
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 to the (much more recent) problem of constructing \emph{dense forests}. These are discrete point sets getting uniformly close to long enough line segments. Progress towards these problems have so far involved a wide range of ideas surrounding areas as varied as combinatorial and computation geometry, convex geometry, Diophantine approximation, discrepancy theory, the theory of dynamical systems, the theory of exponential sums, Fourier analysis, homogeneous dynamics, the mathematical theory of quasicrystals and probability theory. The goal of this paper is to survey the known results related to the Danzer Problem and to the construction of dense forests, to generalise some of them and to state a number of open problems to make further progress towards a solution to this longstanding question.
