If you cut the interval [0,1] into two parts inductively you get classically 2^n boundary points of the sub-intervals at generation n. But if you erase the points of generation n after cutting the sub-intervals of generation n you may get n+1 points at generation n+1. In this talk we study the limit of the empirical distribution of these points when we take different deterministic and random ways of cutting intervals. It is a joint work with James Norris, Michel Pain and Gennady Samorodnitsky.