Abstract
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:V→W 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∈\(\mathcal{P}\).
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[B1] V. Bergelson,Weakly mixing PET, Ergodic Theory Dynam. Systems7 (1987), 337–349.
[B2] V. Bergelson,Ergodic theory and Diophantine problems, inTopics in Symbolic Dynamics and Applications, London Math. Soc. Lecture Note Ser.279 (2000), 167–205.
[BFM] V. Bergelson, H. Furstenberg and R. McCutcheon,IP sets and polynomial recurrence, Ergodic Theory Dynam. Systems16 (1996), 963–974.
[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.
[BL2] V. Bergelson and A. Leibman,Set-polynomials and polynomial extension of the Hales-Jewett theorem, Ann. of Math. (2)150 (1999), 33–75.
[BM] V. Bergelson and R. McCutcheon,Uniformity in polynomial Szemerédi theorem, inErgodic Theory of ℤ d, London Math. Soc. Lecture Note Ser.228 (1996), 273–296.
[BMZ] V. Bergelson, R. McCutcheon and Q. Zhang,A Roth theorem for amenable groups, Amer. J. Math.119 (1997), 1173–1211.
[F1] H. Furstenberg,Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math.31 (1977), 204–256.
[F2] H. Furstenberg,Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, 1981.
[FK1] H. Furstenberg and Y. Katznelson,An ergodic Szemerédi theorem for commuting transformations, J. Analyse Math.34 (1978), 275–291.
[FK2] H. Furstenberg and Y. Katznelson,An ergodic Szemerédi theorem for IP-systems and combinatorial theory, J. Analyse Math.45 (1985), 117–168.
[FK3] H. Furstenberg and Y. Katznelson,A density version of the Hales-Jewett theorem, J. Analyse Math.57 (1991), 64–119.
[G] W. T. Gowers,A new proof of Szemerédi’s theorem, Geom. Funct. Anal.11 (2001), 465–588.
[La] P. Larick, Ph.D. Thesis, The Ohio State University, 1998.
[Le] A. Leibman,Multiple recurrence theorem for measure preserving actions of a nilpotent group, Geom. Funct. Anal.8 (1998), 853–931.
[Sz] E. Szemerédi,On sets of integers containing no k elements in arithmetic progression, Acta Arith.27 (1975), 199–245.
[Z1] R. Zimmer,Extensions of ergodic group actions, Illinois J. Math.20 (1976), 373–409.
[Z2] R. Zimmer,Ergodic actions with generalized discrete spectrum, Illinois J. Math.20 (1976), 555–588.
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The first two authros are supported by NSF, grant DMS-0070566 and DMS-0245350. The second author was supported by the A. Sloan Foundation. The third author is supported by NSF, grant DMS-0200700.
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Bergelson, V., Leibman, A. & McCutcheon, R. Polynomial Szemerédi theorems for countable modules over integral domains and finite fields. J. Anal. Math. 95, 243–296 (2005). https://doi.org/10.1007/BF02791504
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DOI: https://doi.org/10.1007/BF02791504