Encyclopedia of Complexity and Systems Science

2009 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Ergodic Theory: Interactions with Combinatorics and Number Theory

  • Tom Ward
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-30440-3_181

Definition of theSubject

Number theory is a branch of pure mathematicsconcerned with the properties of numbers in general, andintegers in particular. The areas of most relevance to thisarticle are Diophantineanalysis (the study of how real numbers may beapproximated by rational numbers, and the consequences forsolutions of equations in integers); analytic number theory, and inparticular asymptotic estimates for the number of primes smallerthan X asa function of X; equidistribution, andquestions about how the digits of real numbers aredistributed. Combinatorics is concerned with identifyingstructures in discrete objects; of most interest here is thatpart of combinatorics connected with Ramsey theory, assertingthat large subsets of highly structured objects mustautomatically contain large replicas of that structure. Ergodictheory is the study of asymptotic behavior of group actionspreserving a probability measure; it has proved to bea powerful part of dynamical systems with...

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Copyright information

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Authors and Affiliations

  • Tom Ward
    • 1
  1. 1.School of MathematicsUniversity of East AngliaNorwichUK