  We confirm a conjecture of Jens Marklof regarding the equidistribution of
certain sparse collections of points on expanding horospheres. These
collections are obtained by intersecting the expanded horosphere with a certain
manifold of complementary dimension and turns out to be of arithmetic nature.
This equidistribution result is then used along the lines suggested by Marklof
to give an analogue of a result of W. Schmidt regarding the distribution of
shapes of lattices orthogonal to integer vectors.
