# Polynomial Szemerédi theorems for countable modules over integral domains and finite fields

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## Abstract

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* \(\mathcal{P}\) be a finite family of polynomials with integer coefficients and zero constant term. For any α>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∈\(\mathcal{P}\).

## Keywords

Finite Family Formal Degree Finite Dimensional Vector Space Polynomial Versus Ascend Chain Condition## Preview

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