Polynomial szemerédi 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 o φ, where φ: 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 Szemerédi theorem for sets of positive density in finite rank modules over integral domains, as well as the following fact: Let P be a finite family of polynomials with integer coefficients and zero constant term. For any a > 0, there exists N ∈ ℕ such that whenever F is a field with |F| ≥ N and E ⊆ F with |E|/|F| ≥ α, there exist u ∈ F, u ≠ 0, and w ∈ E such that w + φ(u) ∈ E for all φ ∈ P.
Journal d'Analyse Mathematique
Bergelson, V., Leibman, A., & Mccutcheon, R. (2005). Polynomial szemerédi theorems for countable modules over integral domains and finite fields. Journal d'Analyse Mathematique, 95, 243-296. https://doi.org/10.1007/BF02791504