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1 - Ergodic Ramsey Theory

Published online by Cambridge University Press:  30 March 2010

V. Bergelson
Affiliation:
The Ohio State University, Columbus, OH 43210 U.S.A.
Mark Pollicott
Affiliation:
University of Manchester
Klaus Schmidt
Affiliation:
Universität Wien, Austria
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Summary

Introduction.

This survey is an expanded version and elaboration of the material presented by the author at the Workshop on Algebraic and Number Theoretic Aspects of Ergodic Theory which was held in April 1994 as part of the 1993/1994 Warwick Symposium on Dynamics ofZn-actions and their connections with Commutative Algebra, Number Theory and Statistical Mechanics. The leitmotif of this paper is: Ramsey theory and ergodic theory of multiple recurrence are two beautiful, tightly intertwined and mutually perpetuating disciplines. The scope of the survey is mostly limited to Ramsey-theoretical and ergodic questions about Zn-partly because of the proclaimed goals of the Warwick Symposium and partly because of the author's hope that Zn-related combinatorics, number theory and ergodic theory can serve as an ideal lure through which the author's missionary zeal will reach as wide an audience of potential adherents to the subject as possible.

To compensate for the selective neglect of details and for the lack of full generality in some of the proofs, which were imposed by natural time and space limitations, a significant effort was spent on accentuation and motivation of ideas which lead to conjectures and techniques on which the proofs of conjectures hinge.

Here now is a brief description of the content of the five sections constituting the body of this survey. In Section 1 three main principles of Ramsey theory are introduced and their connection with the ergodic theory of multiple recurrence is emphasized.

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Publisher: Cambridge University Press
Print publication year: 1996

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  • Ergodic Ramsey Theory
    • By V. Bergelson, The Ohio State University, Columbus, OH 43210 U.S.A.
  • Edited by Mark Pollicott, University of Manchester, Klaus Schmidt, Universität Wien, Austria
  • Book: Ergodic Theory and Zd Actions
  • Online publication: 30 March 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511662812.002
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  • Ergodic Ramsey Theory
    • By V. Bergelson, The Ohio State University, Columbus, OH 43210 U.S.A.
  • Edited by Mark Pollicott, University of Manchester, Klaus Schmidt, Universität Wien, Austria
  • Book: Ergodic Theory and Zd Actions
  • Online publication: 30 March 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511662812.002
Available formats
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Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Ergodic Ramsey Theory
    • By V. Bergelson, The Ohio State University, Columbus, OH 43210 U.S.A.
  • Edited by Mark Pollicott, University of Manchester, Klaus Schmidt, Universität Wien, Austria
  • Book: Ergodic Theory and Zd Actions
  • Online publication: 30 March 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511662812.002
Available formats
×