A 1965 problem due to Danzer asks whether there exists a set in Euclidean space with finite density intersecting any convex body of volume one. A recent approach to this problem is concerned with the construction of dense forests and is obtained by a suitable weakening of the volume constraint. A dense forest is a discrete point set of finite density getting uniformly close to long enough line segments. The distribution of points in a dense forest is then quantified in terms of a visibility function. Another way to weaken the assumptions in Danzer's problem is by relaxing the density constraint. In this respect, a new concept is introduced in this paper, namely that of an optical forest. An optical forest in $\mathbb{R}^{d}$ is a point set with optimal visibility but not necessarily with finite density. In the literature, the best constructions of Danzer sets and dense forests lack effectivity. The goal of this paper is to provide constructions of dense and optical forests which yield the best known results in any dimension $d \ge 2$ both in terms of visibility and density bounds and effectiveness. Namely, there are three main results in this work: (1) the construction of a dense forest with the best known visibility bound which, furthermore, enjoys the property of being deterministic; (2) the deterministic construction of an optical forest with a density failing to be finite only up to a logarithm and (3) the construction of a planar Peres-type forest (that is, a dense forest obtained from a construction due to Peres) with the best known visibility bound. This is achieved by constructing a deterministic digital sequence satisfying strong dispersion properties.

翻译：由Danzer 引起的1965年问题 由Danzer 引起的问题 问, Euclidean 空间中是否有一组具有有限密度的定点,该空间与第一卷的任何圆形体交织在一起。 这个问题的最近办法涉及密集森林的建造, 其体积限制通过适当的削弱获得。 稠密森林是一组离散的定点, 密度统一接近足够长的线段。 密密森林的点分布随后用可见度功能量化。 削弱Danzer 问题的假设的另一个方法是放宽密度限制。 在这方面, 本文引入了一个新概念, 即光学森林的概念。 $\ mathb{R ⁇ d}$的光学森林是一组最佳可见度但不一定具有有限密度的点。 在文献中, 最佳的丹泽组和稠密森林的构造缺乏效果。 本文的目标是提供密度和光学森林的构造, 仅以任何层面获得最明显的结果 $ge 2 。 。 在这方面, 光学森林的清晰度是三种主要结果: 构建一个最清晰的森林的构造, 以已知的形态为固定的规律 。