In this talk, we mainly focus on dynamical entropy on homogeneous spaces and review the entropy rigidity on homogeneous dynamics following the Einsiedler-Lindenstrauss lecture note (Diagonal actions on locally homogeneous spaces). As its applications, we study the following previous results about Diophantine approximation: Lim-de Saxcé-Shapira (2018) and David-Shapira (2018).

Finally, we introduce the effective version of entropy rigidity and its application to Lim-de Saxcé-Shpira (2018). This is joint work with Wooyeon Kim and Seonhee Lim.