The Sobol' sequence is not quasi-uniform in dimension 2 (Q6566373)

Let \((x_i)_{i\geq 0}\) be a sequence of points in \([0,1]^d\). Assume \(h_n=\sup_{x\in [0,1]^d}\min_{0\leq i\leq n} ||x-x_i||\) and \(q_n=\frac{1}{2} \min_{0\leq i<j\leq n} ||x_i-x_j||\). A sequence \((x_n)\) is called quasi-uniform if there exists a constant \(C>1\) such that \(1\leq \frac{h_n}{q_n}\leq C\) for all \(n\).\N\NIt is proved that the well-known Sobol' sequence is not quasi-uniform in dimension 2.