The case $h=0$ is known as the "Three-Distance Theorem"; just google for numerous references or look [here][1] for discussion and nice pictures, or [here][2] for an interesting historical comment.

A standard reformulation of the theorem is as follows: if, for an irrational $\alpha$, the unit-length circle is partitioned "in the natural way" into $n$ arcs by the points $\alpha k$ with $k\in[1,n]$, then the lengths of these $n$ arcs take just two or three distinct values. This easily implies the case $h\ne 0$ where you basically select one of the arcs (that containing the point corresponding to $h$) and confine to the lengths of the remaining $n-1$ arcs.

[1]:https://sumidiot.wordpress.com/2009/12/23/the-steinhaus-conjecture
[2]:http://www.theoremoftheday.org/NumberTheory/ThreeDistance/TotDThreeDistance.pdf