⨁p∈PFp -Systems as Abramov Systems

Let P be an (unbounded) countable multiset of primes, let G=⨁p∈PFp. We study the k'th universal characteristic factors of an ergodic probability system (X,B,μ) with respect to some measure preserving action of G. We find conditions under which every extension of these factors is generated by phase polynomials and we give an example of an ergodic G-system that is not Abramov. In particular we generalize the main results of Bergelson Tao and Ziegler who proved a similar theorem in the special case P={p,p,p,...} for some fixed prime p. In a subsequent paper we use this result to prove a general structure theorem for ergodic ⨁p∈PFp-systems.