Convergence of measures under diagonal actions on homogeneous spaces

Let λ be a probability measure on Tn−1 where n=2 or 3. Suppose λ is invariant, ergodic and has positive entropy with respect to the linear transformation defined by a hyperbolic matrix. We get a measure μ on SLn(Z)\SLn(R) by putting λ on some unstable horospherical orbit of the right translation of at=diag(et,…,et,e−(n−1)t) (t>0). We prove that if the average of μ with respect to the flow at has a limit, then it must be a scalar multiple of the probability Haar measure. As an application we show that if the entropy of λ is large, then Dirichletʼs theorem is not improvable λ almost surely.