An odd Furstenberg-Szemerédi theorem and quasi-affine systems

Abstract. We prove a version of Furstenberg’s ergodic theorem with restric-tions on return times. More specifically, for a measure preserving system (X,B, µ, T), an integer 0 ≤ j < k, and E ⊂ X with µ(E)> 0, we show that there exists n ≡ j (mod k) with µ(E∩T−nE∩T−2nE∩T−3nE)> 0, so long as T k is ergodic. This result requires a deeper understanding of the limit of some non conventional ergodic averages, and the introduction of a new class of systems, the ‘Quasi-Affine Systems’. 1