Asymptotics of integral points, equivariant compactifications and equidistributions for homogeneous spaces

Let U be a homogeneous variety over Q of a linear algebraic group. Choose an integral model and assume the existence of infinitely many integral points. Then one would like to give an asymptotic count of integral points of bounded height 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 G/H is strongly Hardy-Littlewood with respect to some height function. We also show that when H is ``large'' in G and both G and H are connected, reductive and without nontrivial Q-characters, the asymptotic of integral points is the same as the volume asymptotic up to a constant for every equivariant height. Three concrete examples with explicit heights are also provided to illustrate our approach.