Expanding translates of curves and Dirichlet-Minkowski theorem on linear forms

Abstract

We show that a multiplicative form of Dirichlet's theorem on simultaneous Diophantine approximation as formulated by Minkowski, cannot be improved for almost all points on any analytic curve on R^k which is not contained in a proper affine subspace. Such an investigation was initiated by Davenport and Schmidt in the late sixties. The Diophantine problem is then settled by showing that certain sequence of expanding translates of curves on the homogeneous space of unimodular lattices in R^{k+1} gets equidistributed in the limit. We use Ratner's theorem on unipotent flows, linearization techniques, and a new observation about intertwined linear dynamics of various SL(m,R)'s contained in SL(k+1,R).

28 pages

References in corpus (2)

• Equidistribution of expanding translates of curves and Dirichlet's theorem on Diophantine approximation
• Limiting distributions of curves under geodesic flow on hyperbolic manifolds

Cited by in corpus (7)

• An inhomogeneous Dirichlet theorem via shrinking targets
• Dirichlet's Theorem in function fields
• An Effective Ratner Equidistribution Result for ASL(2,R)
• Hausdorff dimension and uniform exponents in dimension two
• Universal hitting time statistics for integrable flows
• Equidistribution of expanding degenerate manifolds in the space of lattices
• Equidistribution of non-uniformly stretching translates of shrinking smooth curves and weighted Dirichlet approximation