The case $h=0$ is known as the "Three-Distance Theorem"; just google for numerous references or look here for discussion and nice pictures, or here 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.