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 . To prove it, reduce the set modulo 1, and show that all of its members are distinct modulo 1 (show that otherwise would be rational). A consequence is that the set is dense in (0,1), and thus by translation it is dense everywhere.