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 asT ^k 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’. This work was partially carried out while the second author was visiting the Université de Marne la Vallée, supported by NSF grant 9804651.