IPr sets with polynomial weights and Szemerédiʹs theorem

In the mid 1980s H. Furstenberg and Y. Katznelson defined IP r 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 IP r set applications for the density Ramsey theory of abelian groups, including the striking result that, given e > 0 and k ∈ N , there exists some r ∈ N such that for any IP r set R ⊂ Z and any E ⊂ Z with upper density >ϵ, 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.