Encyclopedia of Complexity and Systems Science

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

Ergodic Theory on Homogeneous Spaces and Metric Number Theory

  • Dmitry Kleinbock
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-30440-3_180

Definition of the Subject

The theory of Diophantine approximation , named after Diophantus of Alexandria, in its simplest set-up deals with the approximation of real numbersby rational numbers. Various higher‐dimensional generalizations involve studying values of linear or polynomial maps at integer points. Oftena certain “approximation property” is fixed, and one wants to characterize the set of numbers (vectors, matrices) which share thisproperty, by means of certain measures (Lebesgue, or Hausdorff, or some other interesting measures). This is usually referred to as metric Diophantine approximation .

The starting point for the theory is an elementary fact that ℚ, the set of rational numbers, is dense in ℝ, the reals. In other words,every real number can be approximated by rationals: for any \( { y\in\mathbb{R} } \)
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The work on this paper was supported in part by NSF Grant DMS-0239463.


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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Dmitry Kleinbock
    • 1
  1. 1.Department of MathematicsBrandeis UniversityWalthamUSA