Constructions of Low-discrepancy Sequences - The Van Der Corput Sequence

The Van Der Corput Sequence

Let

$n=\sum_{k=0}^{L-1}d_k(n)b^k$

be the b-ary representation of the positive integer n ≥ 1, i.e. 0 ≤ d_k(n) < b. Set

$g_b(n)=\sum_{k=0}^{L-1}d_k(n)b^{-k-1}.$

Then there is a constant C depending only on b such that (g_b(n))_{n ≥ 1} satisfies

$D^*_N(g_b(1),\dots,g_b(N))\leq C\frac{\log N}{N}.$

where D*_N is the star discrepancy.

Read more about this topic: Constructions Of Low-discrepancy Sequences

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