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  "body": "\n## Description\n\nLet $b \\ge 2$ be an integer.\nEach integer $n \\ge 0$ has a unique base-$b$ expansion $n = \\sum_{k \\ge 0} a_k b^k$ with digits $a_k \\in \\set{0, \\ldots, b-1}$, almost all zero.\nThe *radical inverse* in base $b$ is\n$$\\varphi_b(n) = \\sum_{k \\ge 0} a_k \\, b^{-k-1} \\in [0,1).$$\nIt mirrors the digits of $n$ at the radix point.\nThe sequence $(\\varphi_b(n))_{n \\ge 0}$ is the van der Corput sequence in base $b$.\nLet $s \\ge 1$ and let $b_1, \\ldots, b_s \\ge 2$ be pairwise coprime integers.\nThe *Halton sequence* in these bases is the sequence of points\n$$x_n = \\big(\\varphi_{b_1}(n), \\ldots, \\varphi_{b_s}(n)\\big) \\in [0,1)^s, \\qquad n = 0, 1, 2, \\ldots$$\nThe usual choice is $b_1, \\ldots, b_s$ the first $s$ [[prime numbers|01826KMG2H]].\n\n## Remarks\n\nIn bases $2$ and $3$, the points for $n = 1, 2, 3, 4$ are $(1/2, 1/3)$, $(1/4, 2/3)$, $(3/4, 1/9)$, $(1/8, 4/9)$.\nFor example $3 = 10_3$ in base $3$, so $\\varphi_3(3) = 0.01_3 = 1/9$.\nFor pairwise coprime bases, the star discrepancy of the first $N$ points is $O\\big((\\log N)^s / N\\big)$, with a constant depending on $s$ and the bases.\nSo the Halton sequence is a [[low-discrepancy sequence|XSPX1KW9AQ]], and it is a standard point set for [[quasi-Monte Carlo methods|ABSRB2KXA6]].\nThe coprimality condition matters: with $b_1 = b_2 = 2$, all points lie on the diagonal of the unit square, which they do not fill.\nIn high dimension the coordinates with large bases are correlated for moderate $N$, and scrambled variants are used in practice.\n\n## Explanations\n\n### L1 human\n\nWrite the counting numbers $1, 2, 3, \\ldots$ in base $2$ and read each one backwards after a binary point.\nThis gives $1/2, 1/4, 3/4, 1/8, 5/8, \\ldots$, and each new point falls into one of the largest gaps left by the earlier ones.\nDo the same in base $3$, base $5$, and so on, and use the results as the coordinates of a point.\nThe points fill a square or cube far more evenly than random points do.\nThis makes them good sample points for computing averages.\n",
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