Existence of an optimal dense forest with constant error term

Determine whether there exists a dense forest F ⊂ R^d (for d ≥ 2) with finite density whose visibility satisfies V(ε) = O(ε^{-(d−1)}) as ε → 0; equivalently, ascertain whether the error term E(ε) in the representation V(ε) = ε^{-(d−1)} E(ε^{-1}) can be bounded by a constant (E(ε) ∈ O(1)).

Background

A dense forest is a subset F of R^d with finite density for which there exists a visibility function V(ε) such that any line segment of length V(ε) stays within distance ε of some point of F. Many constructions achieve V(ε) of order ε^{-(d−1)} times a growing error term, with the best bound in this paper achieving a logarithmic error E(ε) ≍ ln(ε^{-1}).

An ‘optimal’ dense forest would have a constant error term E(ε) ∈ O(1), yielding visibility V(ε) = O(ε^{-(d−1)}). Whether such a set exists remains unresolved and represents a central target in minimizing visibility growth.