Equidistributed sequence

Definition

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

lim n → ∞ | { s 1 , … , s n } ∩ [ c , d ] | n = d − c b − a .

(Here, the notation |{s₁,...,s_n }∩[c,d]| denotes the number of elements, out of the first n elements of the sequence, that are between c and d.)

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 {s_n} is a sequence of random variables; rather, it is a determinate sequence of real numbers.

Discrepancy

We define the discrepancy D_N for a sequence {s₁, s₂, s₃, ...} with respect to the interval [a, b] as

D N = sup a ≤ c ≤ d ≤ b | | { s 1 , … , s N } ∩ [ c , d ] | N − d − c b − a | .

A sequence is thus equidistributed if the discrepancy D_N 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. See low-discrepancy sequence for stronger criteria and constructions of low-discrepancy sequences for constructions of sequences which are more evenly distributed.

Riemann integral criterion for equidistribution

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 for an equidistributed sequence:

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

1. The sequence is equidistributed on [a, b].
2. For every Riemann-integrable (complex-valued) function f : [a, b] → C, the following limit holds:

lim N → ∞ 1 N ∑ n = 1 N f ( s n ) = 1 b − a ∫ a b f ( x ) d x

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].

Equidistribution modulo 1

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 a_n, denoted by {a_n} or by a_n−⌊a_n⌋, is equidistributed in the interval [0, 1].

Examples

The equidistribution theorem: The sequence of all multiples of an irrational α,

is equidistributed modulo 1.

More generally, if p is a polynomial with at least one irrational coefficient (other than the constant term) then the sequence p(n) is uniformly distributed modulo 1.

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

The sequence log(n) is not uniformly distributed modulo 1.

The sequence of all multiples of an irrational α by successive prime numbers,

is equidistributed modulo 1. This is a famous theorem of analytic number theory, published by I. M. Vinogradov in 1948.

The van der Corput sequence is equidistributed.

Weyl's criterion

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

lim n → ∞ 1 n ∑ j = 1 n e 2 π i ℓ a j = 0.

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

Generalizations

A quantitative form of Weyl's criterion is given by the Erdős–Turán inequality.

Weyl's criterion extends naturally to higher dimensions, assuming the natural generalization of the definition of equidistribution modulo 1:

The sequence v_n of vectors in R^k is equidistributed modulo 1 if and only if for any non-zero vector ℓ ∈ Z^k,

lim n → ∞ 1 n ∑ j = 0 n − 1 e 2 π i ℓ ⋅ v j = 0.

Example of usage

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.

Suppose α is irrational and denote our sequence by a_j = jα (where j starts from 0, to simplify the formula later). Let ℓ ≠ 0 be an integer. Since α is irrational, ℓα can never be an integer, so e 2 π i ℓ α can never be 1. Using the formula for the sum of a finite geometric series,

| ∑ j = 0 n − 1 e 2 π i ℓ j α | = | ∑ j = 0 n − 1 ( e 2 π i ℓ α ) j | = | 1 − e 2 π i ℓ n α 1 − e 2 π i ℓ α | ≤ 2 | 1 − e 2 π i ℓ α | ,

a 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 is only a finite number of options for the fractional part of a_j = jα.

van der Corput's difference theorem

A theorem of Johannes van der Corput states that if for each h the sequence s_n+h − s_n is uniformly distributed modulo 1, then so is s_n.

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

Metric theorems

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.

For any sequence of distinct integers b_n, the sequence {b_nα} is equidistributed mod 1 for almost all values of α.

The sequence {αⁿ} is equidistributed mod 1 for almost all values of α > 1.

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.

Well-distributed sequence

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

lim n → ∞ | { s k + 1 , … , s k + n } ∩ [ c , d ] | n = d − c b − a

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.

Sequences equidistributed with respect to an arbitrary measure

For an arbitrary probability measure space ( X , μ ) , a sequence of points x n is said to be equidistributed with respect to μ if the mean of point measures converges weakly to μ :

∑ k = 1 n δ x k n ⇒ μ   .

It is true, for example, that for any Borel probability measure on a separable, metrizable space, there exists an equidistributed sequence (with respect to the measure).