  Let U be a homogeneous variety over rational numbers of some linear algebraic
group. Fix an integral model and assume the existence of infinitely many
integral points. Then one would like to give an asymptotic count with the help
of some height function. In many cases, with the help of measure rigidity of
unipotent flows, we reduce this problem to one on equivariant birational
geometry. For instance, we show that if G and H are both connected, semisimple,
simply connected and without compact factors, then U=G/H is strongly
Hardy-Littlewood with respect to certain height function. We also show that
when H is "large" in H and both G and H are connected, reductive and without
non-trivial Q-characters, then for every equivariant height, the asymptotic of
integral points is the same as the volume asymptotic up to a constant. Three
concrete examples with explicit heights are also provided to illustrate our
approach.
