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

Abstract

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.