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An Ergodic IP Polynomial Szemerédi Theorem
Vitaly Bergelson, Ohio State University, Columbus, OH, and Randall McCutcheon, University of Maryland, College Park, MD
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Memoirs of the American Mathematical Society
2000; 106 pp; softcover
Volume: 146
ISBN-10: 0-8218-2657-3
ISBN-13: 978-0-8218-2657-7
List Price: US$49
Individual Members: US$29.40
Institutional Members: US$39.20
Order Code: MEMO/146/695
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We prove a polynomial multiple recurrence theorem for finitely many commuting measure preserving transformations of a probability space, extending a polynomial Szemerédi theorem appearing in [BL1]. The linear case is a consequence of an ergodic IP-Szemerédi theorem of Furstenberg and Katznelson ([FK2]). Several applications to the fine structure of recurrence in ergodic theory are given, some of which involve weakly mixing systems, for which we also prove a multiparameter weakly mixing polynomial ergodic theorem. The techniques and apparatus employed include a polynomialization of an IP structure theory developed in [FK2], an extension of Hindman's theorem due to Milliken and Taylor ([M], [T]), a polynomial version of the Hales-Jewett coloring theorem ([BL2]), and a theorem concerning limits of polynomially generated IP-systems of unitary operators ([BFM]).

Readership

Researchers interested in measure-preserving transformations, partitions of integers, Ramsey theory, sequences and sets.

Table of Contents

  • Introduction
  • Formulation of main theorem
  • Preliminaries
  • Primitive extensions
  • Relative polynomial mixing
  • Completion of the proof
  • Measure-theoretic applications
  • Combinatorial applications
  • For future investigation
  • Appendix: Multiparameter weakly mixing PET
  • References
  • Index of notation
  • Index
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