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

The main goal of this work is to establish quantitative nondivergence estimates for flows on homogeneous spaces of products of real and p-adic Lie groups. These results have applications both to

Systems of small linear forms and Diophantine approximation on manifolds

We develop the theory of Diophantine approximation for systems of simultaneously small linear forms, which coefficients are drawn from any given analytic non-degenerate manifolds. This setup

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



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

In this paper we discuss homogeneous Diophantine approximation of points on smooth manifolds M in ℝk. We begin with a brief survey of the notation and results. For any x,y ∈ℝk, let .

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