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An Ergodic IP Polynomial Szemerédi Theorem

Vitaly Bergelson Ohio State University, Columbus, OH
Randall McCutcheon University of Maryland, College Park, MD
Available Formats:
Electronic ISBN: 978-1-4704-0286-0
Product Code: MEMO/146/695.E
106 pp
List Price: $52.00 MAA Member Price:$46.80
AMS Member Price: $31.20 Click above image for expanded view An Ergodic IP Polynomial Szemerédi Theorem Vitaly Bergelson Ohio State University, Columbus, OH Randall McCutcheon University of Maryland, College Park, MD Available Formats:  Electronic ISBN: 978-1-4704-0286-0 Product Code: MEMO/146/695.E 106 pp  List Price:$52.00 MAA Member Price: $46.80 AMS Member Price:$31.20
• Book Details

Memoirs of the American Mathematical Society
Volume: 1462000
MSC: Primary 28; Secondary 05; 11;

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]).

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

• Chapters
• 0. Introduction
• 1. Formulation of main theorem
• 2. Preliminaries
• 3. Primitive extensions
• 4. Relative polynomial mixing
• 5. Completion of the proof
• 6. Measure-theoretic applications
• 7. Combinatorial applications
• 8. For future investigation
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Volume: 1462000
MSC: Primary 28; Secondary 05; 11;

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]).

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

• Chapters
• 0. Introduction
• 1. Formulation of main theorem
• 2. Preliminaries
• 3. Primitive extensions
• 4. Relative polynomial mixing
• 5. Completion of the proof
• 6. Measure-theoretic applications
• 7. Combinatorial applications
• 8. For future investigation
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