I start with a noncompact connected semisimple Lie group with finite center $G$ and fix a maximal compact subgroup $K$ of $G$. I am considering these compact groups $K$. If $\mathbb T$ is the maximal torus in $K$, I take the quotient $K/\mathbb T$. I read that they are Kahler manifold. I am interested to know if there is a natural way to find a dense set in $K/\mathbb T$ which is parametrized by $(j_1/2^{n_1}, \dots, j_m/2^{n_m})$, where $j_i\in \mathbb Z$ and $n_i\in \mathbb N$.

Something like a dense grid, as can be constructed on $\mathbb R^d$ and some other groups/spaces like in a Heisenberg group, is what I have in mind.