The content of the talk is a joint work with Elon Lindenstrauss. Let $X$ be the space of unimodular (covolume 1) lattices in Euclidean d-space and let $A$ denote the group of diagonal matrices of determinant $1$. We prove that any lattice $x$ in $X$ which "comes from a number field" which is not a CM field satisfies a Ratner-like property, namely the closure of the orbit $Ax$ equals to an orbit $Hx$ of a group $H$ containing $A$. As a consequence we generalize my previous work on Diophantine properties of totally real cubic numbers by dropping the dimension assumption and the totally realness.