Equidistributed sequence

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In mathematics, a bounded sequence {s_n}_{n = 1, 2, 3, ...} of real numbers is equidistributed on an interval (a, b) precisely if for any subinterval (c, d) 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. the proportion of terms falling in any subinterval is proportional to the length of the subinterval. Loosely speaking one could say that each point in the sequence is equally likely to fall anywhere in its range. However, this is not about sequences of random variables; we contemplate sequences of constants with no randomness.

In analytic number theory, the fractional parts of the sequences

$\vartheta, 2\vartheta, 3\vartheta,\dots$

and

$2\vartheta, 3\vartheta, 5\vartheta, \dots$

are equidistributed in the interval $[0,1)$ for all irrational $\vartheta$ (where the coefficients in the second sequence are the primes), the latter result being a famous theorem of I. M. Vinogradov.

[edit] Equidistribution mod 1

In mathematics, a sequence { a_n : n = 1, 2, 3, ... } is equidistributed modulo 1 precisely if for every interval (a, b) within the larger interval [0, 1),

$\lim_{n\to\infty} {\left| \{\, a_1-\lfloor a_1 \rfloor, \dots, a_n-\lfloor a_n \rfloor \,\} \cap (a,b) \right| \over n} = b-a.$

In other words, the long-run proportion of fractional parts of a_n that fall within any subinterval is just the length of the subinterval. For example, since the interval (0.5, 0.8) occupies 30% of the space within the larger interval (0, 1), the proportion of members of the sequence whose fractional part falls between 0.5 and 0.8 will approach 0.3.

The equidistribution theorem states that for any irrational number a, the sequence a, 2a, 3a, ... is equidistributed mod 1. A powerful general result is Weyl's criterion, which shows that equidistribution is equivalent to having a non-trivial estimate for the exponential sums formed with the sequence as exponents. For example, it reduces this case of the multiples of a to summing finite geometric series.

Equidistributions are studied in Diophantine approximation theory.