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Ergodic Theory on Homogeneous Spaces and Metric Number Theory

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Encyclopedia of Complexity and Systems Science

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 \( { y\in\mathbb{R} } \) and any \( { \varepsilon > 0 } \) there exists \( { p/q \in\mathbb{Q} } \) with

$$ |y -p/q| < \varepsilon\:. $$
(1)

To answer questions like “how well can various real...

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Abbreviations

Diophantine approximation :

Diophantine approximation refers to approximation of real numbers by rational numbers, or more generally, finding integer points at which some (possibly vector‐valued) functions attain values close to integers.

Metric number theory :

Metric number theory (or, specifically, metric Diophantine approximation) refers to the study of sets of real numbers or vectors with prescribed Diophantine approximation properties.

Homogeneous spaces:

A homogeneous space \( { G/\Gamma } \) of a group G by its subgroup Γ is the space of cosets \( { \{g\Gamma\} } \). When G is a Lie group and Γ is a discrete subgroup, the space \( { G/\Gamma } \) is a smooth manifold and locally looks like G itself.

Lattice; unimodular lattice :

A lattice in a Lie group is a discrete subgroup of finite covolume; unimodular stands for covolume equal to 1.

Ergodic theory :

The study of statistical properties of orbits in abstract models of dynamical systems.

Hausdorff dimension :

A nonnegative number attached to a metric space and extending the notion of topological dimension of “sufficiently regular” sets, such as smooth submanifolds of real Euclidean spaces.

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Acknowledgment

The work on this paper was supported in part by NSF Grant DMS-0239463.

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  1. Department of Mathematics, Brandeis University, Waltham, USA

    Dmitry Kleinbock

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  1. Dmitry Kleinbock
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Kleinbock, D. (2009). Ergodic Theory on Homogeneous Spaces and Metric Number Theory. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_180

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