Call a vector X = (x, y) in R2 "resonant" if it satisfies an equation
of the form ax + by + c = 0 where a, b, c are integers, not all zero.
Otherwise X is non-resonant. Use Baire theory to show that the set
of "non-resonant" vectors is dense in R2.
All I have so far is that a vector is non-resonant if it satisfies x not rational, y not rational or x /= q1 + q2*y, where q1 and q1 are rational.
Note: Apply Baire's theory is a nice way of saying that we have to show the set is thick; that is, the intersection of countably many open and dense sets.