  Saturating sets are combinatorial objects in projective spaces over finite
fields that have been intensively investigated in the last three decades. They
are related to the so-called covering problem of codes in the Hamming metric.
In this paper, we consider the recently introduced linear version of such sets,
which is, in turn, related to the covering problem in the rank metric. The main
questions in this context are how small the rank of a saturating linear set can
be and how to construct saturating linear sets of small rank. Recently, Bonini,
Borello, and Byrne provided a lower bound on the rank of saturating linear sets
in a given projective space, which is shown to be tight in some cases. In this
paper, we provide construction of saturating linear sets meeting the lower
bound and we develop a link between the saturating property and the
scatteredness of linear sets. The last part of the paper is devoted to show
some parameters for which the bound is not tight.
