Uniform distribution of prime powers and sets of recurrence and van der Corput sets in ℤ k

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 Sárkőzy, Furstenberg, Kamae and Mèndes France, and Bergelson and Lesigne. The proofs utilize a general equidistribution result involving prime powers which is of independent interest.