IPr sets with polynomial weights and Szemeredi's theorem

yIn 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 Szemeredi theorem for commuting groups of measure preserving transformations, many IP, set applications for the density Ramsey theory of abelian groups, including the striking result that, given e > 0 and k is an element of N, there exists some r is an element of N such that for any IPr set R subset of Z and any E subset of Z with upper density > epsilon, E contains a k-term arithmetic progression having common difference r is an element of R. Here, polynomial versions of these results are obtained as applications of a recently proved polynomial extension to the Furstenberg-Katznelson IP Szemeredi theorem. (c) 2006 Elsevier Inc. All rights reserved.