Journal d’Analyse Mathématique

, Volume 95, Issue 1, pp 243–296 | Cite as

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

  • V. Bergelson
  • A. Leibman
  • R. McCutcheon


Given a pair of vector spacesV andW over a countable fieldF and a probability spaceX, one defines apolynomial measure preserving action ofV onX to be a compositionT o ϕ, where ϕ:VW is a polynomial mapping andT is a measure preserving action ofW onX. 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∈\(\mathcal{P}\).


Finite Family Formal Degree Finite Dimensional Vector Space Polynomial Versus Ascend Chain Condition 
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Copyright information

©  0251 V 2 2005

Authors and Affiliations

  1. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA
  2. 2.Department of Mathematical SciencesUniversity of MemphisMemphisUSA

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