Equidistributed sequence

In mathematics, a bounded sequence {s_{1}, s_{2}, s_{3}, …} 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 interval. Such sequences are studied in Diophantine approximation theory and have applications to Monte Carlo integration.
Contents
Definition
A bounded sequence {s_{1}, s_{2}, s_{3}, …} of real numbers is said to be equidistributed on an interval [a, b] if for any subinterval [c, d] of [a, b] we have
(Here, the notation {s_{1},…,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 {s_{1}, s_{2}, s_{3}, …} with respect to the interval [a, b] as
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 lowdiscrepancy sequence for stronger criteria and constructions of lowdiscrepancy sequences for constructions of sequences which are more evenly distributed.
Equidistribution modulo 1
The sequence {a_{1}, a_{2}, a_{3}, …} is said to be equidistributed modulo 1 or uniformly distributed modulo 1 if the sequence of the fractional parts of the a_{n}'s, (denoted by a_{n}−⌊a_{n}⌋)
is equidistributed in the interval [0, 1].
Examples
 The sequence of all multiples of an irrational α,

 0, α, 2α, 3α, 4α, …
is uniformly distributed modulo 1: this is the equidistribution theorem.
 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 proved by Weyl and is an application of the theorem of Johannes van der Corput.
 The sequence log(n) is not uniformly distributed modulo 1.
 The sequence of all multiples of an irrational α by successive prime numbers,

 2α, 3α, 5α, 7α, 11α, …
is equidistributed modulo 1. This is a famous theorem of analytic number theory, proved by I. M. Vinogradov in 1935.
 The van der Corput sequence is equidistributed.
Properties
The following three conditions are equivalent:
 {a_{n}} is equidistributed modulo 1.
 For every Riemann integrable function f on [0, 1],
 For every nonzero integer k,

The third condition is known as Weyl's criterion. Together with the formula for the sum of a finite geometric series, the equivalence of the first and third conditions furnishes an immediate proof of the equidistribution theorem.
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_{n} α} is equidistributed mod 1 for almost all values of α.^{[1]}
 The sequence {α^{n}} is equidistributed mod 1 for almost all values of α > 1.^{[2]}
It is not known whether the sequences {e^{n}} or {π^{n}} are equidistributed mod 1. However it is known that the sequence {α^{n}} is not equidistributed mod 1 if α is a PV number.
Welldistributed sequence
A bounded sequence {s_{1}, s_{2}, s_{3}, …} of real numbers is said to be welldistributed on [a, b] if for any subinterval [c, d] of [a, b] we have
uniformly in k. Clearly every welldistributed sequence is uniformly distributed, but the converse does not hold. The definition of welldistributed modulo 1 is analogous.
See also
References
 ^ See Satz 1, Über eine Anwendung der Mengenlehre auf ein aus der Theorie der säkularen Störungen herrührendes Problem, Felix Bernstein, Mathematische Annalen 71, #3 (September 1911), pp. 417439, doi:10.1007/BF01456856.
 ^ Ein mengentheoretischer Satz über die Gleichverteilung modulo Eins, J. F. Koksma, Compositio Mathematica, 2 (1935), pp. 250258.
 L. Kuipers; H. Niederreiter (2006). Uniform Distribution of Sequences. Dover Publishing. ISBN 0486450198.
 L. Kuipers; H. Niederreiter (1974). Uniform Distribution of Sequences. John Wiley & Sons Inc.. ISBN 0471510459.
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