Number Theory pp 447-492 | Cite as

Uniform Distribution and Ergodic Theory

  • W. A. Coppel
Part of the Universitext book series (UTX)


A trajectory of a system which is evolving with time may be said to be ‘recurrent’ if it keeps returning to any neighbourhood, however small, of its initial point, and ‘dense’ if it passes arbitrarily near to every point. It may be said to be ‘uniformly distributed’ if the proportion of time it spends in any region tends asymptotically to the ratio of the volume of that region to the volume of the whole space. In the present chapter these notions will be made precise and some fundamental properties derived. The subject of dynamical systems has its roots in mechanics, but we will be particularly concerned with its applications in number theory.


Positive Integer Ergodic Theory Ergodic Theorem Arithmetic Progression Nonzero Vector 
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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • W. A. Coppel
    • 1
  1. 1.GriffithAustralia

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