  We introduce and study bi-Lipschitz-invariant dimensions that range between
the box and Assouad dimensions. The quasi-Assouad dimensions and
$\theta$-spectrum are other special examples of these intermediate dimensions.
These dimensions are localized, like Assouad dimensions, but vary in the depth
of scale which is considered, thus they provide very refined geometric
information.
  We investigate the relationship between these and the familiar dimensions. We
construct a Cantor set with a non-trivial interval of dimensions, the endpoints
of this interval being given by the quasi-Assouad and Assouad dimensions of the
set. We study continuity-like properties of the dimensions. In contrast with
the Assouad-type dimensions, we see that decreasing sets in $\mathbb{R}$ with
decreasing gaps need not have dimension $0$ or $1$.
  Formulas are given for the dimensions of Cantor-like sets and these are used
in some of our constructions. We also show that, as is the case for Hausdorff
and Assouad dimensions, the Cantor set and the decreasing set have the extreme
dimensions among all compact sets in $\mathbb{R}$ whose complementary set
consists of open intervals of the same lengths.
