Equidistributed sequence

In mathematics, a sequence (s1, s2, s3, ...) of real numbers is said to be equidistributed, or uniformly distributed, if the proportion of terms falling in a subinterval is proportional to the length of that subinterval. Such sequences are studied in Diophantine approximation theory and have applications to Monte Carlo integration.

A sequence (s1, s2, s3, ...) of real numbers is said to be equidistributed on a non-degenerate interval [a, b] if for any subinterval [c, d] of [a, b] we have

For example, if a sequence is equidistributed in [0, 2], since the interval [0.5, 0.9] occupies 1/5 of the length of the interval [0, 2], as n becomes large, the proportion of the first n members of the sequence which fall between 0.5 and 0.9 must approach 1/5. Loosely speaking, one could say that each member of the sequence is equally likely to fall anywhere in its range. However, this is not to say that (sn) is a sequence of random variables; rather, it is a determinate sequence of real numbers.

We define the discrepancy DN for a sequence (s1, s2, s3, ...) with respect to the interval [ab] as

A sequence is thus equidistributed if the discrepancy DN tends to zero as N tends to infinity.

Equidistribution is a rather weak criterion to express the fact that a sequence fills the segment leaving no gaps. For example, the drawings of a random variable uniform over a segment will be equidistributed in the segment, but there will be large gaps compared to a sequence which first enumerates multiples of ε in the segment, for some small ε, in an appropriately chosen way, and then continues to do this for smaller and smaller values of ε. For stronger criteria and for constructions of sequences that are more evenly distributed, see low-discrepancy sequence.

Recall that if f is a function having a Riemann integral in the interval [a, b], then its integral is the limit of Riemann sums taken by sampling the function f in a set of points chosen from a fine partition of the interval. Therefore, if some sequence is equidistributed in [a, b], it is expected that this sequence can be used to calculate the integral of a Riemann-integrable function. This leads to the following criterion[1] for an equidistributed sequence:

Suppose (s1, s2, s3, ...) is a sequence contained in the interval [a, b]. Then the following conditions are equivalent:

This criterion leads to the idea of Monte-Carlo integration, where integrals are computed by sampling the function over a sequence of random variables equidistributed in the interval.

It is not possible to generalize the integral criterion to a class of functions bigger than just the Riemann-integrable ones. For example, if the Lebesgue integral is considered and f is taken to be in L1, then this criterion fails. As a counterexample, take f to be the indicator function of some equidistributed sequence. Then in the criterion, the left hand side is always 1, whereas the right hand side is zero, because the sequence is countable, so f is zero almost everywhere.

In fact, the de Bruijn–Post Theorem states the converse of the above criterion: If f is a function such that the criterion above holds for any equidistributed sequence in [a, b], then f is Riemann-integrable in [a, b].[2]

A sequence (a1, a2, a3, ...) of real numbers is said to be equidistributed modulo 1 or uniformly distributed modulo 1 if the sequence of the fractional parts of an, denoted by (an) or by an − ⌊an⌋, is equidistributed in the interval [0, 1].

This was proven by Weyl and is an application of van der Corput's difference theorem.[4]

Weyl's criterion states that the sequence an is equidistributed modulo 1 if and only if for all non-zero integers ℓ,

The criterion is named after, and was first formulated by, Hermann Weyl.[7] It allows equidistribution questions to be reduced to bounds on exponential sums, a fundamental and general method.

The sequence vn of vectors in Rk is equidistributed modulo 1 if and only if for any non-zero vector ℓ ∈ Zk,

Weyl's criterion can be used to easily prove the equidistribution theorem, stating that the sequence of multiples 0, α, 2α, 3α, ... of some real number α is equidistributed modulo 1 if and only if α is irrational.[3]

a finite bound that does not depend on n. Therefore, after dividing by n and letting n tend to infinity, the left hand side tends to zero, and Weyl's criterion is satisfied.

Conversely, notice that if α is rational then this sequence is not equidistributed modulo 1, because there are only a finite number of options for the fractional part of aj = .

A theorem of Johannes van der Corput[8] states that if for each h the sequence sn+h − sn is uniformly distributed modulo 1, then so is sn.[9][10][11]

A van der Corput set is a set H of integers such that if for each h in H the sequence sn+h − sn is uniformly distributed modulo 1, then so is sn.[10][11]

Metric theorems describe the behaviour of a parametrised sequence for almost all values of some parameter α: that is, for values of α not lying in some exceptional set of Lebesgue measure zero.

It is not known whether the sequences (en) or (πn) are equidistributed mod 1. However it is known that the sequence (αn) is not equidistributed mod 1 if α is a PV number.

A sequence (s1, s2, s3, ...) of real numbers is said to be well-distributed on [a, b] if for any subinterval [c, d] of [a, b] we have

uniformly in k. Clearly every well-distributed sequence is uniformly distributed, but the converse does not hold. The definition of well-distributed modulo 1 is analogous.

In any Borel probability measure on a separable, metrizable space, there exists an equidistributed sequence with respect to the measure; indeed, this follows immediately from the fact that such a space is standard.

The general phenomenon of equidistribution comes up a lot for dynamical systems associated with Lie groups, for example in Margulis' solution to the Oppenheim conjecture.