A New Construction of Forests with Low Visibility

A set of points with finite density is constructed in ${\mathbb{R}}^d$, with $d\ge 2$, by adding points to a Poisson process such that any line segment of length $O({\epsilon }^{-(d-1)}\ln {\epsilon }^{-1})$ in ${\mathbb{R}}^d$ will contain one of the points of the set within distance $\epsilon$ of it. The constant implied by the big-$O$ notation depends on the dimension only.