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).

Publication Title

Israel Journal of Mathematics