Let $x$ be an irrational number, and $\beta$ strictly larger than its irrationality index, which means that for some $C>0$, for all $n\in \mathbb{Z}^*$, $$d(nx,\mathbb{Z})>C n^{-\beta}.$$

Are there some ways to obtain bounds on the linear combination of two irrational numbers $x,y$? That is, for $n,m\in \mathbb{Z}^*$, by what could I bound from below $$d(nx+my,\mathbb{Z})?$$ I also assume that $x/y$ is also irrational.

EDIT: I am interested by any number $x,y$ such that something can be said.