On Rationally Ergodic and Rationally Weakly Mixing Rank-One   Transformations
  We study the notions of weak rational ergodicity and rational weak mixing as defined by Jon Aaronson. We prove that various families of infinite measure-preserving rank-one transformations possess (or do not posses) these properties, and consider their relation to other notions of mixing in infinite measure. 
Irving Dai and Xavier Garcia and Tudor P\u{a}durariu and Cesar E.   Silva
math.DS