Flows on homogeneous spaces and Diophantine approximation on manifolds

  title={Flows on homogeneous spaces and Diophantine approximation on manifolds},
  author={Dmitry Kleinbock and G. A. Margulis},
  journal={Annals of Mathematics},
We present a new approach to metric Diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces. This approach yields a new proof of a conjecture of Mahler, originally settled by V. G. Sprindzuk in 1964. We also prove several related hypotheses of Baker and Sprindzuk formulated in 1970s. The core of the proof is a theorem which generalizes and sharpens earlier results on nondivergence… 
Flows on S-arithmetic homogeneous spaces and applications to metric Diophantine approximation
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An inhomogeneous transference principle and Diophantine approximation
In a landmark paper (‘Flows on homogeneous spaces and Diophantine approximation on manifolds’, Ann. of Math. (2) 148 (1998), 339–360.) Kleinbock and Margulis established the fundamental
A Groshev Type Theorem for Convergence on Manifolds
We deal with Diophantine approximation on the so-called non-degenerate manifolds and prove an analogue of the Khintchine–Groshev theorem. The problem we consider was first posed by A. Baker [1] for
Inhomogeneous theory of dual Diophantine approximation on manifolds
Equidistribution of expanding translates of curves and Dirichlet’s theorem on diophantine approximation
We show that for almost all points on any analytic curve on ℝk which is not contained in a proper affine subspace, the Dirichlet’s theorem on simultaneous approximation, as well as its dual result
Exceptional Sets in Dynamical Systems and Diophantine Approximation
The nature and origin of exceptional sets associated with the rotation number of circle maps, Kolmogorov-Arnol’d-Moser theory on the existence of invariant tori and the linearisation of complex
Diophantine Properties of Measures and Homogeneous Dynamics
This is a survey of the so-called “quantitative nondivergence” approach to metric Diophantine approximation developed approximately 10 years ago in my collaboration with Margulis. The goal of this
Federer Measures, Good and Nonplanar Functions of Metric Diophantine Approximation
The goal of this paper is to generalize the main results of [1] and subsequent papers on metric Diophantine approximation with dependent quantities to the set-up of systems of linear forms. In
S-arithmetic inhomogeneous Diophantine approximation on manifolds


Flows on homogeneous spaces and Diophantine properties of matrices
We generalize the notions of badly approximable (resp. singular) systems of m linear forms in n variables, and relate these generalizations to certain bounded (resp. divergent) trajectories in the
Metric Diophantine approximation and Hausdorff dimension on manifolds
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On orbits of unipotent flows on homogeneous spaces
  • S. Dani
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 1984
Abstract Let G be a connected Lie group and let Γ be a lattice in G (not necessarily co-compact). We show that if (ut) is a unipotent one-parameter subgroup of G then every ergodic invariant (locally
Divergent trajectories of flows on homogeneous spaces and Diophantine approximation.
Let G be a connected Lie group and Γ be a lattice in G; that is, Γ is a discrete subgroup of G such that G/Γ admits a finite G-invariant measure. Let {gt}teP be a oneparameter subgroup of G. The
Contributions to the theory of transcendental numbers
This volume consists of a collection of papers devoted primarily to transcendental number theory and diophantine approximations written by the author. Most of the materials included in this volume
Discrete subgroups of Lie groups
Preliminaries.- I. Generalities on Lattices.- II. Lattices in Nilpotent Lie Groups.- III. Lattices in Solvable Lie Groups.- IV. Polycyclic Groups and Arithmeticity of Lattices in Solvable Lie
Transcendental Number Theory
First published in 1975, this classic book gives a systematic account of transcendental number theory, that is, the theory of those numbers that cannot be expressed as the roots of algebraic
ContentsIntroduction I. Metrical theory of approximation on manifolds ??1. The basic problem ??2. Brief survey of results ??3. The principal conjecture II. Metrical theory of transcendental numbers
Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability
Acknowledgements Basic notation Introduction 1. General measure theory 2. Covering and differentiation 3. Invariant measures 4. Hausdorff measures and dimension 5. Other measures and dimensions 6.
Diophantine approximation in ⁿ
The diophantine approximation deals with the approximation of real numbers (or real vectors) with rational numbers (or rational vectors). See the article Wikipedia article Diophantine_approximation