Host-Kra theory for-systems and multiple recurrence

Let be an (unbounded) countable multiset of primes (that is, every prime may appear multiple times) and let. We develop a Host-Kra structure theory for the universal characteristic factors of an ergodic G-system. More specifically, we generalize the main results of Bergelson, Tao and Ziegler [An inverse theorem for the uniformity seminorms associated with the action of. Geom. Funct. Anal. 19(6) (2010), 1539-1596], who studied these factors in the special case for some fixed prime p. As an application we deduce a Khintchine-Type recurrence theorem in the flavor of Bergelson, Tao and Ziegler [Multiple recurrence and convergence results associated to-Actions. J. Anal. Math. 127 (2015), 329-378] and Bergelson, Host and Kra [Multiple recurrence and nilsequences. Invent. Math. 160(2) (2005), 261-303, with an appendix by I. Ruzsa].