I am given a set of $M$ points in a closed interval.
I would like to partition the segment (with equidistant points), in such a way that my partition contains all these chosen points and since there are infinitely many such partitions, I want to keep the coarsest one.
Is this easy? I don't see how to do it at once.
Pick one end of the line segment as the starting point. Each point $x_i$ of the $M$ points is some from the start of the line segment; let $d_i$ be the ratio of the distance of $x_i$ from the start of the line segment to the total distance of the line segment. If any of the $d_i$ are not rational numbers, you cannot divide up your line segment equally in the way you want to. If they are all rational numbers, let $\frac{p_i}{q_i}$ be $d_i$ in lowest terms. Then you will want to divide up your line segment into $lcm(q_1, q_2, \ldots, q_m)$ equal pieces to get your coarsest possible partition.