The Sobol’ sequence is not quasi-uniform in dimension 2

Abstract

Are common quasi-Monte Carlo sequences quasi-uniform? While this question remains widely open, in this short note, we prove that the two-dimensional Sobol’ sequence is not quasi-uniform. This result partially answers an unsolved problem of Sobol’ and Shukhman [Math. Comput. Simulation 75 (2007), pp. 80–86] in a negative manner.