**Memoirs of the American Mathematical Society**

2000;
106 pp;
Softcover

MSC: Primary 28;
Secondary 05; 11

Print ISBN: 978-0-8218-2657-7

Product Code: MEMO/146/695

List Price: $52.00

Individual Member Price: $31.20

**Electronic ISBN: 978-1-4704-0286-0
Product Code: MEMO/146/695.E**

List Price: $52.00

Individual Member Price: $31.20

# 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

#### Readership

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