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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?

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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.

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