We extend the main result of [V. Bergelson, I. Kornfeld, and B. Mityag
in, Proc. Amer. Math. Sec. 119 (1993), 1127-1134.] to more general com
mutative group actions satisfying different mixing conditions. The mor
e general setup required introduction of new techniques based on spect
ral theory as well as refinement of those used in that paper. (C) 1994
Academic Press, Inc.