Equidistribution of expanding translates of curves and Dirichlet’s theorem on diophantine approximation

By Nimish A. Shah. 2008

59 citations recorded by the source corpus.

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 for simultaneous approximation of linear forms, cannot be improved. The result is obtained by proving asymptotic equidistribution of evolution of a curve on a strongly unstable leaf under certain partially hyperbolic flow on the space of unimodular lattices in ℝk+1. The proof involves Ratner’s theorem on ergodic properties of unipotent flows on homogeneous spaces.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/19c9b80991004376db4f11ef83bd3fa82432eb99

Citations in the corpus

• On effective equidistribution of expanding translates of certain orbits in the space of lattices (2007, unknown): “That is, gt-translates of H-orbits become equidistributed asbtc!1 uniformly in z when the latter is restricted to compact subsets of X. The proof relies on S . G. Dani’s classication of measures invariant under horospherical subgroups and the so-called ‘linearization method’.”
• Metric Diophantine Approximation : aspects of recent work (2016, unknown): “By exploiting the work of Shah [87], it is shown in [55] that for any non-degenerate planar curve C ∣∣∣ C ∩ ⋂ γ∈R2W × γ ∣∣∣ C = 1 .”