In the divergence case of Khintchine's theorem, Schmidt obtained an asymptotic formula for the number of rational approximations of bounded height to almost every real number. Using exponential mixing in the space of lattices, we prove versions of this theorem for intrinsic diophantine approximation on grassmannians.

For every vector $\overline \alpha\in \RR^n$ and for every rational approximation $(\overline p,q)\in \RR^n\times\RR$ we can associate the displacement vector $q\alpha-\overline p$.
We focus on algebraic vectors, namely $\overline \alpha=(\alpha_1,\dots,\alpha_n)$ such that $1, \alpha_1, \dots, \alpha_n$ span a rank $n$ number field.
For these vectors, we will discuss the size of their displacements as well as the distribution of their directions.
We will discuss a new proof for the p-adic littlewood conjecture for the algebraic vectors, and in addition, exploit this proof to classify all limiting distributions, with a special weighting, of the sequence of directions of the displacements in the $\varepsilon$-approximations of $(\alpha_1, \dots, \alpha_n)$. Each such limiting measure will be expressed as the pushforward of an algebraic measure on $X_n$ to the sphere.