Low-discrepancy sequence

Remarks

Example: the van der Corput sequence in base $2$, with $s = 1$, is $x_i = \sum_k a_k 2^{-k-1}$ when $i = \sum_k a_k 2^k$ in binary; its first terms are $\frac12, \frac14, \frac34, \frac18$. It satisfies $D_N^* = O(\log N / N)$ in big O notation, so it is a low-discrepancy sequence. Non-example: the constant sequence $x_i = 0$ has $A_N(t) = N$ for every $t$ with all $t_j > 0$, so $D_N^* = 1$ for all $N$, and the bound fails. A sequence has $D_N^* \to 0$ if and only if it is uniformly distributed in $[0, 1)^s$; for $s = 1$ this is equidistribution modulo one. So every low-discrepancy sequence is uniformly distributed, but the converse fails. For independent uniform random points, $D_N^*$ is typically of order $N^{-1/2}$ up to logarithmic factors, much larger than $(\log N)^s / N$ for large $N$. Some authors require the bound only for $N$ large, or use the extreme discrepancy over all boxes, not only those anchored at $0$; this changes only the constant.