Let $G$ be an abelian subgroup of $SL(d,Z)$. When $G$ acts totally irreducibly on $T^d$ the $d$-dimensional torus, has some hyperbolicity and is not virtually-cyclic, Berend proved that every orbit on $T^d$ is either the whole torus or finite. We will discuss effective forms of this theorem and how they are related to number-theoretical problems. This is an analogue of the recent quantitative Furstenberg's theorem concerning the $\times 2, \times 3$ action on the circle by Bourgain-Lindenstrauss-Michel-Venkatesh.