Littlewood's conjecture is a long-standing open problem in Diophantine approximation, and is closely related to the (higher rank) action of diagonal matrices on the space of lattices in $\mathbb{R}^3$. In the previous work, I established some quantitative result on Littlewood's conjecture, by using the measure rigidity property of the diagonal action and studying the behavior of empirical measures. Recently, I improved this result by refining the method of the proof, and, in this talk, I will explain about this improvement.