# An Ergodic IP Polynomial Szemerédi Theorem

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*Vitaly Bergelson; Randall McCutcheon*

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]).

#### Table of Contents

# Table of Contents

## An Ergodic IP Polynomial Szemeredi Theorem

- Contents vii8 free
- Abstract viii9 free
- 0. Introduction 110 free
- 1. Formulation of main theorem 1019 free
- 2. Preliminaries 1524
- 3. Primitive Extensions 2635
- 4. Relative polynomial mixing 3746
- 5. Completion of the proof 5564
- 6. Measure-theoretic applications 6473
- 7. Combinatorial applications 8190
- 8. For future investigation 9099
- A. Appendix: Multiparameter weakly mixing PET 92101
- References 103112
- Index of Notation 105114 free
- Index 106115