nt.number theory - Gaps in nx (mod 1)

This reminds me of the following cute statement (which I believe has been recently generalized to arbitrary manifolds, but I couldn't find the paper): Given the sequence of points $kalpha (mod 1)$, there are at most 3 different distances between nearest neighbors.

Actually, I first heard it stated as $alpha^k$ on the unit circle. To prove it, notice that the first cycle of points around the circle all have the same length between them, and then there is the remainder at the end.

When the wrap occurs, this is the remainder just shifted (because the distance between two consecutive points is always the same), and it keeps appearing between previous points in the same way at each step (and then there are the remaining points it has yet to appear between). This still leaves only 3 lengths (the two from the "cut" it produced in the previous length, and the previous lengths it has yet to cut).

I hope this helps.