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Approximately Transitive Actions of Abelian Groups and Spectrum

It is proved that any measure-preserving approximately transitive (AT) action of a locally compact separable Abelian group on a Lebesgue space has simple spectrum. The proof is based on investigation of properties of nontran-sient product cocycles and actions associated with them. We used our result that two nontransient cocycles with isomorphic associated (Mackey) actions are weakly equivalent.