  We introduce diagonal dimension, a version of nuclear dimension for diagonal
sub-C*-algebras (sometimes also referred to as diagonal C*-pairs). Our concept
has good permanence properties and detects more refined information than
nuclear dimension. In many situations it is precisely how dynamical information
is encoded in an associated C*-pair.
  For free actions on compact Hausdorff spaces, diagonal dimension of the
crossed product with its canonical diagonal is bounded above by a product
involving Kerr's tower dimension of the action and covering dimension of the
space. It is bounded below by the dimension of the space, by the asymptotic
dimension of the group, and by the fine tower dimension of the action. For a
locally compact, Hausdorff, \'etale groupoid, diagonal dimension of the
groupoid C*-algebra is bounded below by the dynamic asymptotic dimension of the
groupoid. For free Cantor dynamical systems, diagonal dimension (defined at the
level of the crossed product C*-algebra) and tower dimension (an entirely
dynamical notion) agree on the nose. Similarly, for a finitely generated group
diagonal dimension of its uniform Roe algebra with the canonical diagonal
agrees precisely with asymptotic dimension of the group. This statement also
holds for uniformly bounded metric spaces. We apply the lower bounds above to a
number of further examples which show how diagonal dimension keeps track of
information not seen by nuclear dimension.
