RANK-ONE WEAK MIXING FOR NONSINGULAR TRANSFORMATIONS

We construct rank-one infinite measure preserving transformations sati sfying each of the following dynamical properties: (1) Continuous L-in finity spectrum, conservative k-fold cartesian products but nonergodic cartesian square; (2) ergodic k-fold cartesian products; (3) nonconse rvative cartesian square. We show how to modify the construction of (1 ) to obtain type IIIlambda transformations with similar properties.