in this talk, we discuss effective versions of ratner’s theorems in the space of affine lattices. for $d \geq 2$, let $y=asl_d(\mathbb{r})/asl_d(\mathbb{z})$, $h$ be a minimal horospherical group of $sl_d(\mathbb{r})$ embedded in $asl_d(\mathbb{r})$, and $a_t$ be the corresponding diagonal flow. then $(a_t)$-push-forwards of a piece of $h$-orbit become equidistributed with a polynomial error rate under certain diophantine condition of the initial point of the orbit. this generalizes the previous results of strömbergsson for $d = 2$ and of prinyasart for $d = 3$.