
Constructions of low-discrepancy sequences
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The van der Corput sequence
Let
be the b-ary representation of the positive integer n ≥ 1, i.e. 0 ≤ dk(n) < b. Set

Then there is a constant C depending only on b such that (gb(n))n ≥ 1 satisfies

where D*N is the
star discrepancy.
The Halton sequence
The Halton sequence is a natural generalization of the van der Corput sequence to higher dimensions. Let s be an arbitrary dimension and b1, ..., bs be arbitrary coprimeCoprime
In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...
integers greater than 1. Define

Then there is a constant C depending only on b1, ..., bs, such that sequence {x(n)}n≥1 is a s-dimensional sequence with

The Hammersley set
Let b1,...,bs-1 be coprimeCoprime
In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...
positive integers greater than 1. For given s and N, the s-dimensional Hammersley
John Hammersley
John Michael Hammersley was a British mathematician best known for his foundational work in the theory of self-avoiding walks and percolation theory. He was born in Helensburgh in Dunbartonshire, and educated at Sedbergh School. He started reading mathematics at Emmanuel College, Cambridge but was...
set of size N is defined by

for n = 1, ..., N. Then

where C is a constant depending only on b1, ..., bs−1.