On the uniform distribution modulo 1 of multidimensional LS-sequences

Ingrid Carbone introduced the notion of so-called LS-sequences of points, which are obtained by a generalization of Kakutani’s interval splitting procedure. Under an appropriate choice of the parameters L and S, such sequences have low discrepancy, which means that they are natural candidates for Quasi-Monte Carlo integration. It is tempting to assume that LS-sequences can be combined coordinatewise to obtain a multidimensional low-discrepancy sequence. However, in the present paper, we prove that this is not always the case: if the parameters L1,S1 and L2,S2 of two one-dimensional low-discrepancy LS-sequences satisfy certain number-theoretic conditions, then their two-dimensional combination is not even dense in [0,1]2.