Hi, I want to know whether the set S={a+bt|a,b are rational integers} is discrete, where 0<t<1 is an irrational number. Namely, does the set S intersect each interval (-r,r) with only finite elements? Thank you very much!
No, the set is dense in $\displaystyle \mathbb{R}$. To prove it, reduce the set $\displaystyle \{bt : b \in \mathbb{Z}\}$ modulo 1, and show that all of its members are distinct modulo 1 (show that otherwise $\displaystyle t$ would be rational). A consequence is that the set is dense in (0,1), and thus by translation it is dense everywhere.