Polynomial Szemeredi theorems for countable modules over integral domains and finite fields

Given a pair of vector spaces V and W over a countable field F and a probability space X, one defines a polynomial measure preserving action of V on X to be a composition T circle rho, where rho : V -> W is a polynomial mapping and T is a measure preserving action of W on X. We show that the known structure theory of measure preserving group actions extends to polynomial actions and establish a Furstenberg-style multiple recurrence theorem for such actions. Among the combinatorial corollaries of this result are a polynomial Szemeredi theorem for sets of positive density in finite rank modules over integral domains, as well as the following fact: Let T be a finite family of polynomials with integer coefficients and zero constant term. For any alpha > 0, there exists N is an element of N such that whenever F is afield with vertical bar F vertical bar >= N and E subset of F with vertical bar E vertical bar/vertical bar F vertical bar >= alpha, there exist u is an element of F, u not equal 0, and w is an element of E such that w + rho(u) is an element of E for all rho is an element of P.