On the Corner Avoidance Properties of Various Low-discrepancy Sequences

The minimum distance of QMC sample points to the boundary of the unit cube is an important quantity in the error analysis of QMC integration for functions with singularities. Sobol’ and recently Owen show that the Sobol’ and Halton sequences avoid a hyperbolically shaped region around the corners of the unit cube. We extend these results in two ways. First, we prove that generalized Niederreiter sequences possess similar properties as Sobol’ and Halton sequences around the origin. Second, we show corner avoidance rates for the Halton and Faure sequence for corners different from the origin. While the all-corner avoidance of the Halton sequence is almost the same as its origin avoidance, the Faure sequence has a substantially smaller all-corner avoidance.