Discrepancy of LS-sequences of partitions and points

In this paper, we study a countable family of uniformly distributed sequences of partitions, called LS-sequences of partitions, and we give a precise estimate of their discrepancy. Among these sequences, we identify a countable class having low discrepancy (which means of order ${{\frac{1}{N}}}$ ). We describe an explicit algorithm that associates to each of these sequences a uniformly distributed sequence of points (we call LS-sequences of points). The main result of this paper says that the discrepancy of the sequences of points associated by our algorithm to the LS-sequences of partitions is of order α _N log N, if α _N is the discrepancy of the corresponding sequence of partitions. We obtain therefore, in particular, a countable family of low-discrepancy sequences of points.