I would not consider dealing with a unit circle to be a different method. Considering a circle vs numbers mod 1 is just a reformulation of the problem.

This is indeed a classic problem. A stronger statement that $\displaystyle \{a_i(x)\}$ is not only dense but uniform is

the equidistribution theorem. The original fact follows from the observation that $\displaystyle \inf\{m\cdot1+n\cdot x\mid m\cdot1+n\cdot x>0\mbox{ and }m,n\in\mathbb{Z}\}=0$ since this set does not have a minimum; otherwise, the minimum would be a common divisor of 1 and x and they would be commensurate. (Of course, the absence of a minimum by itself does not imply that the $\displaystyle \inf$ is 0; another simple step is needed.) See also

this thread and

this post. See also

this statement on cut-the-knot (set a = b = 0). The fact also follows from the

Dirichlet's approximation theorem.