Limiting distribution of expanding translates of sparse set of points on certain horosphere

The orbit of certain expanding horosphere of dimension $mn$ from the identity coset in $SL(m+n,R)/SL(n+m,Z)$ intersects the orbit of its opposite and contracting horospherical in a dense set of rational points. We consider certain finite primitive collection of points of denominator $k$ in this collection and translate them by a diagonal element expanding these points at certain rate depending on $k$, and describe their limit distributions as $k$ tends to infinity depending on the rate of expansion. This a joint work with M. Einsiedler, S. Mozes and U. Shapira.