Uniform Distribution of Prime Powers and sets of Recurrence and van der Corput sets in Z^k

Abstract: We establish new results on sets of recurrence and van der Corput sets in Z^k which refine and unify some of the previous results obtained by Sarkozy, Furstenberg, Kamae and Mendes France, and Bergelson and Lesigne. The proofs utilize a general equidistribution result involving prime powers which is of independent interest.