EQUIDISTRIBUTION OF EXPANDING CURVES IN HOMOGENEOUS SPACES AND DIOPHANTINE APPROXIMATION ON SQUARE MATRICES

摘要

In this paper, we study an analytic curve phi : I = [a, b]. -> M(n x n, R) in the space of n by n real matrices. There is a natural map u : M(n x n, R) -> H = SL(2n, R). Let G be a Lie group containing H and Gamma < G be a lattice of G. Let X = G/Gamma. Then given a dense H-orbit in X, one could embed u(phi(I)) into X. We consider the expanding translates of the curve by some diagonal subgroup A = {a(t) : t is an element of R} subset of H. We will prove that if phi satisfies certain geometric conditions, then the expanding translates will tend to be equidistributed in G/Gamma, as t -> + infinity. As an application, we show that for almost every point on phi(I), the Diophantine approximation given by Dirichlet's Theorem is not improvable.