  A dense forest is a set $F \subset \mathbb{R}^n$ with the property that for
all $\varepsilon > 0$ there exists a number $V(\varepsilon) > 0$ such that all
line segments of length $V(\varepsilon)$ are $\varepsilon$-close to a point in
$F$. The function $V$ is called a visibility function of $F$. In this paper we
study dense forests constructed from finite unions of translated lattices
(grids). First, we provide a necessary and sufficient condition for a finite
union of grids to be a dense forest in terms of the irrationality properties of
the matrices defining them. This answers a question raised by Adiceam, Solomon,
and Weiss (2022). To complement this, we further show that such sets
generically admit effective visibility bounds in the following sense: for all
$\eta > 0$, there exists a $k \in \mathbb{N}$ such that almost all unions of
$k$ grids are dense forests admitting a visibility function $V(\varepsilon) \ll
\varepsilon^{-(n-1) -\eta}$. This is arbitrarily close to optimal in the sense
that if a finite union of grids admits a visibility function $V$, then this
function necessarily satisfies $V(\varepsilon) \gg \varepsilon^{-(n-1)}$. One
of the main novelties of this work is that the notion of `almost all' is
considered with respect to several underlying measures, which are defined
according to the Iwasawa decomposition of the matrices used to define the
grids. In this respect, the results obtained here vastly extend those of
Adiceam, Solomon, and Weiss (2022) who provided similar effective visibility
bounds for a particular family of generic unimodular lattices.
