# linear combinations of of irrational number with integer coefficients.

Let $$\{x_1, \dots,x_k\}$$ be real numbers, under which conditions do I have that the set $$\langle x_1, \dots,x_k\rangle = \{\sum a_i x_i: a_i \in \mathbb{Z} \}$$ is discrete set?

Well, this is clearly the case if $$x_i$$'s are all integers. On the other hand, if I take $$x_1=1$$ and $$x_2$$ to be the Liouville Constant, I have a dense set. So what would be necessary and/or sufficient conditions?

If $$\frac x y$$ is irrational then $$\{nx+my:n,m \in \mathbb Z\}$$ is dense. If $$\frac x y$$ is rational then $$\{nx+my:n,m \in \mathbb Z\}$$ is discrete.