equidistribution

Equidistribution describes a property of a sequence of points becoming uniformly spread over a space in the limit. For a sequence x_n in a compact space X with a probability measure μ, the sequence is equidistributed if the empirical measures (1/N) ∑ δ_{x_n} converge weakly to μ. In the unit interval with Lebesgue measure, a sequence (x_n) is equidistributed modulo 1 if, for every subinterval [a,b] ⊆ [0,1], the proportion of indices n ≤ N with {x_n} ∈ [a,b] tends to b−a as N → ∞.

Weyl's criterion provides a practical test: a sequence (x_n) is equidistributed mod 1 if and only if, for every nonzero integer h, the averages (1/N) ∑ e^{2π i h x_n} tend to 0. This criterion allows proving equidistribution for many sequences and is a central tool in the theory.

Classic examples include the sequence (nα) mod 1 with α irrational, which is equidistributed. More generally, Weyl's theorem asserts that the sequence {P(n)} is equidistributed mod 1 whenever P is a real polynomial with at least one irrational coefficient in positive degree. Equidistribution also extends to higher-dimensional tori, where sequences in R^d modulo Z^d are required to have uniform distribution in the d-dimensional unit cube.

Applications appear in analytic number theory, ergodic theory, and numerical methods. Equidistributed sequences underpin uniform sampling in quasi-Monte Carlo methods and provide insight into the distribution of fractional parts of sequences.