The van der Corput sequence

Let


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


Then there is a constant C depending only on b such that (g_b(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, ..., b_s be arbitrary coprime

Coprime

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, ..., b_s, such that sequence {x(n)}_n≥1 is a s-dimensional sequence with

The Hammersley set

Let b₁,...,b_s-1 be coprime

Coprime

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, ..., b_s−1.