Let be a line segment of length broken into pieces of length at most . It’s easy to break into blocks (using the preexisting breakpoints) that differ in length by at most (break at the nearest available point to , etc.).
In the case where each piece has length and the number of pieces isn’t divisible by , we can’t possibly do better than a maximum difference of between blocks. Is this achievable in general?
In today’s combinatorics seminar Imre Bárány described joint work with Victor Grinberg which says that the answer is yes.