Equidistributed

Equidistributed describes a distribution property of a sequence of real numbers with respect to the unit interval or another measure, indicating that the sequence spreads out evenly in the limit. In the common setting, a sequence (x_n) is equidistributed modulo 1 if the fractional parts {x_n} become uniformly distributed in the interval [0, 1).

Formally, for every subinterval [a, b) ⊆ [0, 1], the limit as N → ∞ of (1/N) times the number of indices n ≤ N with {x_n} ∈ [a, b) equals b − a. Equivalently, the empirical distribution of the fractional parts converges to the uniform distribution on [0, 1]. A related notion is equidistribution with respect to a probability measure μ on [0, 1].

Weyl's criterion provides a practical test: the sequence {x_n} is equidistributed modulo 1 if and only if for every nonzero integer h, the limit (1/N) ∑_{n=1}^N e^{2π i h x_n} = 0 as N → ∞. This often reduces checking equidistribution to analyzing exponential sums.

A standard example is the sequence {nα} modulo 1. If α is irrational, the sequence is equidistributed in [0, 1], while if α is rational, it takes only finitely many fractional values and is not equidistributed.

In higher dimensions, a sequence in R^d is equidistributed modulo 1 if its coordinates are jointly equidistributed on the d-torus [0, 1)^d. Equidistribution underlies many results in number theory and numerical methods, including quasi-Monte Carlo integration and the study of discrepancy.