Equidistributed sequence

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In mathematics, a bounded sequence (s_n)_{n = 1, 2, 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

$\lim_{n\to\infty}{ \left|\{\,s_1,\dots,s_n \,\} \cap [c,d] \right| \over n}={d-c \over b-a},$

i.e., if the proportion of terms falling in any subinterval is proportional to the length of the subinterval. 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.

[edit] Equidistribution modulo 1

The sequence (a_n)_{n = 1, 2, 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, ( $a_n-\lfloor a_n\rfloor$ )_{n = 1, 2, 3, ...}, is equidistributed in the interval [0, 1].

For any irrational α, the sequence of all multiples of α,

0, α, 2α, 3α, 4α, …

and the sequence of all multiples of α by a prime number,

2α, 3α, 5α, 7α, 11α, …

are both equidistributed modulo 1. The first of these results is called the equidistribution theorem; the second is a famous theorem of analytic number theory, proved by I. M. Vinogradov in 1935.

The following three conditions are equivalent:

(a_n) is equidistributed modulo 1.
For every Riemann integrable function f on [0, 1],

$\lim_{n\to\infty} \frac{1}{n} \sum_{1\le j\le n} f(a_j)=\int_0^1 f(x)\, dx.$

For every nonzero integer k,

$\lim_{n\to\infty} \frac{1}{n} \sum_{1\le j\le n} e^{2\pi ik a_j}=0.$

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.

Equidistribution is studied in Diophantine approximation theory.