Weyl sequence

In mathematics, a Weyl sequence is a sequence from the equidistribution theorem proven by Hermann Weyl:^[1]

The sequence of all multiples of an irrational α,

0, α, 2α, 3α, 4α, ...

is equidistributed modulo 1.^[2]

In other words, the sequence of the fractional parts of each term will be uniformly distributed in the interval [0, 1).

In computing, an integer version of this sequence is often used to generate a discrete uniform distribution rather than a continuous one. Instead of using an irrational number, which cannot be calculated on a digital computer, the ratio of two integers is used in its place. An integer k is chosen, relatively prime to an integer modulus m. In the common case that m is a power of 2, this amounts to requiring that k is odd.

The sequence of all multiples of such an integer k,

0, k, 2k, 3k, 4k, …

is equidistributed modulo m.

That is, the sequence of the remainders of each term when divided by m will be uniformly distributed in the interval [0, m).

The term appears to originate with George Marsaglia’s paper "Xorshift RNGs".^[3] The following C code generates what Marsaglia calls a "Weyl sequence":

d += 362437;

In this case, the odd integer is 362437, and the results are computed modulo m = 2³² because d is a 32-bit quantity. The results are equidistributed modulo 2³².

• List of things named after Hermann Weyl

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