FVIP systems and multiple recurrence
An IP system is a function n taking finite subsets of N to a commutative, additive group Ω satisfying n(α ∪ β) = n(α) + n(β) whenever α ∩ β= θ. In an extension of their Szemerédi theorem for finitely many commuting measure preserving transformations, Furstenberg and Katznelson showed that if Si, 1 ≤ i ≤ k, are IP systems into a commutative (possibly infinitely generated) group n of measure preserving transformations of a probability space (X, B, μ), and A ∈ B with μ(A) > 0, then for some θ ≠ α one has μ(∩i=1k(α)A) > 0. We extend this to so-called FVIP systems, which are polynomial analogs of IP systems, thereby generalizing as well joint work by the author and V. Bergelson concerning special FVIP systems of the form S(α) = T(p(n(α))), where p: Z t → Zd is a polynomial vanishing at zero, T is a measure preserving Zd action and n is an IP system into Z t. The primary novelty here is potential infinite generation of the underlying group action, however there are new applications in Zd as well, for example multiple recurrence along a wide class of generalized polynomials (very roughly, functions built out of regular polynomials by iterated use of the greatest integer function).
Israel Journal of Mathematics
Mccutcheon, R. (2005). FVIP systems and multiple recurrence. Israel Journal of Mathematics, 146, 157-188. https://doi.org/10.1007/BF02773532