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  "body": "\n## Definition\n\nLet $s \\ge 1$ be an integer, and let $x_1, x_2, \\ldots$ be a sequence of points of $[0, 1)^s$.\nFor $t = (t_1, \\ldots, t_s) \\in [0, 1]^s$ let $A_N(t)$ be the number of indices $i \\le N$ with $x_i \\in [0, t_1) \\times \\cdots \\times [0, t_s)$.\nThe *star discrepancy* of the first $N$ points is\n$$D_N^* = \\sup_{t \\in [0, 1]^s} \\left| \\frac{A_N(t)}{N} - t_1 t_2 \\cdots t_s \\right|.$$\nThe sequence is a *low-discrepancy sequence* if some constant $c$, independent of $N$, satisfies\n$$D_N^* \\le c\\, \\frac{(\\log N)^s}{N} \\qquad \\text{for all } N \\ge 2.$$\n\n## Remarks\n\nExample: 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 $\\frac12, \\frac14, \\frac34, \\frac18$.\nIt satisfies $D_N^* = O(\\log N / N)$ in [[big O notation|0TGV80E6A4]], so it is a low-discrepancy sequence.\nNon-example: the constant sequence $x_i = 0$ has $A_N(t) = N$ for every $t$ with all $t_j \u003e 0$, so $D_N^* = 1$ for all $N$, and the bound fails.\nA 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|XM6DH57BWY]].\nSo every low-discrepancy sequence is uniformly distributed, but the converse fails.\nFor 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 $N$.\nSome 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.\n",
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