An odd Furstenberg-Szemerédi theorem and quasi-affine systems

We prove a version of Furstenberg’s ergodic theorem with restrictions on return times. More specifically, for a measure preserving system (X, B, μ,T), integers 0 ≤j <k, andE ⊂X with μ(E) > 0, we show that there existsn ≡ j (modk) with ώ(E ∩T-nE ∩T-2nE ∩T-3nE) > 0, so long asTk is ergodic. This result requires a deeper understanding of the limit of some nonconventional ergodic averages and the introduction of a new class of systems, the ‘Quasi-Affine Systems’.