  By previous work of Giordano and the author, ergodic actions of $\Z$ (and
other discrete groups) are completely classified measure-theoretically by their
dimension space, a construction analogous to the dimension group used in
C*-algebras and topological dynamics. Here we investigate how far from AT
(approximately transitive) can actions be which derive from circulant (and
related) matrices. It turns out not very: although non-AT actions can arise
from this method of construction, under very modest additional conditions,
ATness arises; in addition, if we drop the positivity requirement in the
isomorphism of dimension spaces, then all these ergodic actions satisfy an
analogue of AT. Many examples are provided
