Danzer's Problem, Effective Constructions of Dense Forests and Digital Sequences

Abstract: 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.