  Let $V$ be an $r$-dimensional $\mathbb{F}_{q^n}$-vector space. We call an
$\mathbb{F}_q$-subspace $U$ of $V$ $h$-scattered if $U$ meets the
$h$-dimensional $\mathbb{F}_{q^n}$-subspaces of $V$ in $\mathbb{F}_q$-subspaces
of dimension at most $h$. In 2000 Blokhuis and Lavrauw proved that
$\dim_{\mathbb{F}_q} U \leq rn/2$ when $U$ is $1$-scattered. Subspaces
attaining this bound have been investigated intensively because of their
relations with projective two-weight codes and strongly regular graphs.
MRD-codes with a maximum idealiser have also been linked to $rn/2$-dimensional
$1$-scattered subspaces and to $n$-dimensional $(r-1)$-scattered subspaces.
  In this paper we prove the upper bound $rn/(h+1)$ for the dimension of
$h$-scattered subspaces, $h>1$, and construct examples with this dimension. We
study their intersection numbers with hyperplanes, introduce a duality relation
among them, and study the equivalence problem of the corresponding linear sets.
