  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.
