A new isomorphism invariant of certain measure preserving flows, using sequences of integers, is introduced. Using this invariant, we are able to construct large families of type III0 systems which are not orbit equivalent. In particular we construct an uncountable family of nonsingular ergodic transformations, each having an associated flow that is approximately transitive (and therefore of zero entropy), with the property that the transformations are pairwise not orbit equivalent.
Dooley, A. H., Hawkins, J., & Ralston, D. (2011). Families of type III0 ergodic transformations in distinct orbit equivalent classes. Monatshefte fur Mathematik, 164(4), 369-381. https://doi.org/10.1007/s00605-010-0258-0