The aim of this talk is to present a recent joint work with Professor Gregory A. Margulis. Our main result is that any indefinite ternary integral quadratic form is integrally equivalent to many forms whose coefficients are small. This is deduced from a concentration property of the periodic SO(1, 2)-orbits in SL(3, R)/SL(3,Z).

This will be a case study of a particular theme in homogeneous dynamics which will highlight connections with Diophantine approximation. I will start with a historical survey and then will mention some new work, joint with Jinpeng An and Lifan Guan