# 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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## References

- [B1] V. Bergelson,
*Weakly mixing PET*, Ergodic Theory Dynam. Systems**7**(1987), 337–349.MATHMathSciNetCrossRefGoogle Scholar - [B2] V. Bergelson,
*Ergodic theory and Diophantine problems*, in*Topics in Symbolic Dynamics and Applications*, London Math. Soc. Lecture Note Ser.**279**(2000), 167–205.Google Scholar - [BFM] V. Bergelson, H. Furstenberg and R. McCutcheon,
*IP sets and polynomial recurrence*, Ergodic Theory Dynam. Systems**16**(1996), 963–974.MATHMathSciNetCrossRefGoogle Scholar - [BL1] V. Bergelson and A. Leibman,
*Polynomial extensions of van der Waerden's and Szemerédi's theorems*, J. Amer. Math. Soc.**9**(1996), 725–753.MATHCrossRefMathSciNetGoogle Scholar - [BL2] V. Bergelson and A. Leibman,
*Set-polynomials and polynomial extension of the Hales-Jewett theorem*, Ann. of Math. (2)**150**(1999), 33–75.MATHCrossRefMathSciNetGoogle Scholar - [BM] V. Bergelson and R. McCutcheon,
*Uniformity in polynomial Szemerédi theorem*, in*Ergodic Theory of ℤ*^{d}, London Math. Soc. Lecture Note Ser.**228**(1996), 273–296.Google Scholar - [BMZ] V. Bergelson, R. McCutcheon and Q. Zhang,
*A Roth theorem for amenable groups*, Amer. J. Math.**119**(1997), 1173–1211.MATHCrossRefMathSciNetGoogle Scholar - [F1] H. Furstenberg,
*Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions*, J. Analyse Math.**31**(1977), 204–256.MATHMathSciNetGoogle Scholar - [F2] H. Furstenberg,
*Recurrence in Ergodic Theory and Combinatorial Number Theory*, Princeton University Press, 1981.Google Scholar - [FK1] H. Furstenberg and Y. Katznelson,
*An ergodic Szemerédi theorem for commuting transformations*, J. Analyse Math.**34**(1978), 275–291.MATHMathSciNetGoogle Scholar - [FK2] H. Furstenberg and Y. Katznelson,
*An ergodic Szemerédi theorem for IP-systems and combinatorial theory*, J. Analyse Math.**45**(1985), 117–168.MATHMathSciNetGoogle Scholar - [FK3] H. Furstenberg and Y. Katznelson,
*A density version of the Hales-Jewett theorem*, J. Analyse Math.**57**(1991), 64–119.MATHMathSciNetGoogle Scholar - [G] W. T. Gowers,
*A new proof of Szemerédi’s theorem*, Geom. Funct. Anal.**11**(2001), 465–588.MATHCrossRefMathSciNetGoogle Scholar - [La] P. Larick, Ph.D. Thesis, The Ohio State University, 1998.Google Scholar
- [Le] A. Leibman,
*Multiple recurrence theorem for measure preserving actions of a nilpotent group*, Geom. Funct. Anal.**8**(1998), 853–931.MATHCrossRefMathSciNetGoogle Scholar - [Sz] E. Szemerédi,
*On sets of integers containing no k elements in arithmetic progression*, Acta Arith.**27**(1975), 199–245.MATHMathSciNetGoogle Scholar - [Z1] R. Zimmer,
*Extensions of ergodic group actions*, Illinois J. Math.**20**(1976), 373–409.MATHMathSciNetGoogle Scholar - [Z2] R. Zimmer,
*Ergodic actions with generalized discrete spectrum*, Illinois J. Math.**20**(1976), 555–588.MATHMathSciNetGoogle Scholar