  Linnik proved in the late 1950's the equidistribution of integer points on
large spheres under a congruence condition. The congruence condition was lifted
in 1988 by Duke (building on a break-through by Iwaniec) using completely
different techniques. We conjecture that this equidistribution result also
extends to the pairs consisting of a vector on the sphere and the shape of the
lattice in its orthogonal complement. We use a joining result for higher rank
diagonalizable actions to obtain this conjecture under an additional congruence
condition.
