We consider the action of the diagonal subgroup {a(t)=(tn−1,t−1,…,t−1)}⊂G=SL(n,R) on X=G/Γ, where Γ=SL(n,Z). Let C be a finite piece of an analytic curve on the expanding horophere (≅Rn−1) of {a(t)}t>1 in G . Let μC be a smooth probability measure on the trajectory C[Γ] on X. We provide necessary and sufficient conditions on the smallest affine subspace containing C in terms of Diophantine approximation and algebraic number fields so that the measures a(t)μC get equidistributed in X as t→∞. This result generalizes the speaker’s earlier work showing equidistribution of translates of curves, which are not contained in proper affine subspaces. The result answers a question of Davenport and Schmidt on non-improvability of Dirichlet’s approximation. The case of n=3 is a joint work D. Kleinbock, N. de Saxcé, and P. Yang; and the general case is a joint work with Pengyu Yang.