Recurrence and primitivity for IP systems with polynomial wildcards

The IP Szemerédi Theorem of Furstenberg and Katznelson guarantees that for any positive density subset E of a countable abelian group G and for any sequences (Formula Presented), there is a finite non-empty α ⊂ N such that (Formula Presented). A natural question is whether, in this theorem, one may restrict |α| to, for example, the set (Formula Presented). As a first step toward achieving this result, we develop here a new method for taking weak IP limits and prove a relevant projection theorem for unitary operators, which establishes as a corollary the case k = 2 of the target result