IPr sets with polynomial weights and Szemerédi\u27s theorem

In the mid 1980s H. Furstenberg and Y. Katznelson defined IPr sets in abelian groups as, roughly, sets consisting of all finite sums of r fixed elements. They obtained, via their powerful IP Szemerédi theorem for commuting groups of measure preserving transformations, many IPr set applications for the density Ramsey theory of abelian groups, including the striking result that, given e \u3e 0 and k ∈ N, there exists some r ∈ N such that for any IPr set R ⊂ Z and any E ⊂ Z with upper density \u3eε{lunate}, E contains a k-term arithmetic progression having common difference r ∈ R. Here, polynomial versions of these results are obtained as applications of a recently proved polynomial extension to the Furstenberg-Katznelson IP Szemerédi theorem. © 2006 Elsevier Inc. All rights reserved