Sum-free sets in abelian groups |
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Authors: | Email author" target="_blank">Ben?GreenEmail author Imre?Z?Ruzsa |
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Institution: | (1) Department of Mathematical Sciences, University of Memphis, 38152 Memphis, TN, USA |
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Abstract: | An IP system is a functionn taking finite subsets ofN to a commutative, additive group Ω satisfyingn(α∪β)=n(α)+n(β) whenever α∩β=ø. In an extension of their Szemerédi theorem for finitely many commuting measure preserving transformations, Furstenberg and Katznelson showed that ifS i ,1≤i≤k, are IP systems into a commutative (possibly infinitely generated) group Ω of measure preserving transformations of a probability space (X, B, μ, andA∈B with μ(A)>0, then for some ø≠α one has μ(? i=1 k S i({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 formS(α)=T(p(n(α))), wherep:Z t →Z d is a polynomial vanishing at zero,T is a measure preservingZ d action andn is an IP system intoZ t . The primary novelty here is potential infinite generation of the underlying group action, however there are new applications inZ d as well, for example multiple recurrence along a wide class ofgeneralized polynomials (very roughly, functions built out of regular polynomials by iterated use of the greatest integer function). |
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