For which $\alpha$ do the $\epsilon$-neighborhoods of $\{k\alpha \mod 1 \mid k = 1, \ldots , poly(1/\epsilon) \}$ cover $[0,1]$?
In this paper on quantum computing (last paragraph of page 25), Dorit Aharonov claims
...we require that the sequence $\alpha$ mod 1, 2$\alpha$ mod 1, 3$\alpha$ mod 1, ... hits the $\epsilon$-neighborhood of any number in [0, 1], within $poly(1/\epsilon)$ steps. Clearly, $\alpha$ should be irrational, but not all irrational numbers satisfy this property. It is not very difficult to see that an irrational root of a polynomial of degree 2 satisfies the required property.
However, I still can't see how to see this, especially the part with roots of a quadratic polynomial.