  We revisit the visible points of a lattice in Euclidean $n$-space together
with their generalisations, the $k$th-power-free points of a lattice, and study
the corresponding dynamical system that arises via the closure of the lattice
translation orbit. Our analysis extends previous results obtained by Sarnak and
by Cellarosi and Sinai for the special case of square-free integers and sheds
new light on previous joint work with Peter Pleasants.
