Editing Low-discrepancy sequence (section)

===Hammersley set===

[[File:Hammersley set 2D.svg|thumb|right|2D Hammersley set of size 256]]
Let <math>b_1,\ldots,b_{s-1}</math> be [[coprime]] positive integers greater than 1. For given <math>s</math> and <math>N</math>, the <math>s</math>-dimensional [[John Hammersley|Hammersley]] set of size <math>N</math> is defined by<ref name="HammersleyHandscomb1964">{{cite book|title=Monte Carlo Methods|last1=Hammersley|first1=J. M.|last2=Handscomb|first2=D. C.|year=1964|doi=10.1007/978-94-009-5819-7|isbn=978-94-009-5821-0 }}</ref>

:<math>
x(n)=\left(g_{b_1}(n),\dots,g_{b_{s-1}}(n),\frac{n}{N}\right)
</math>

for <math>n = 1, \ldots, N</math>. Then

:<math>
D^*_N(x(1),\dots,x(N))\leq C\frac{(\log N)^{s-1}}{N}
</math>

where <math>C</math> is a constant depending only on <math>b_1, \ldots, b_{s-1}</math>.

''Note'': The formulas show that the Hammersley set is actually the Halton sequence, but we get one more dimension for free by adding a linear sweep. This is only possible if  <math>N</math> is known upfront. A linear set is also the set with lowest possible one-dimensional discrepancy in general. Unfortunately, for higher dimensions, no such "discrepancy record sets" are known. For <math>s=2</math>, most low-discrepancy point set generators deliver at least near-optimum discrepancies.