Equidistribution theorem

Weyl’s theorem states that for every irrational number α, the fractional parts of nα are uniformly distributed in [0,1) as n ranges over the positive integers.

Weyl criterion: A sequence is uniformly distributed modulo one exactly when its nonzero integer exponential averages tend to zero. It gives the principal analytic test behind Weyl’s theorem.
Fractional part: For a real number x, its fractional part is x minus its greatest integer not exceeding x. The theorem tracks the fractional parts of nα.
Hermann Weyl: A German mathematician whose work shaped mathematical physics, representation theory, and number theory. His 1916 work introduced the theorem and criterion bearing his name.
Irrational rotation: The circle map that repeatedly adds a fixed irrational angle modulo one. Its orbit is exactly the sequence of fractional parts in Weyl’s theorem.
Weyl's equidistribution theorem for polynomials: A theorem stating that polynomial values modulo one are uniformly distributed when at least one nonconstant coefficient is irrational. It extends the linear sequence nα to polynomial phases.
Exponential sum: A finite sum of complex exponentials whose phases encode arithmetic or analytic information. The theorem is proved by showing particular exponential sums become small.
Uniform distribution modulo one: A sequence is uniformly distributed modulo one when its terms visit every interval with its length as limiting frequency. This is the precise property Weyl’s theorem establishes.
Analytic number theory: The branch of number theory that uses techniques from mathematical analysis. Weyl’s theorem exemplifies analytic methods applied to arithmetic sequences.
Benford's law: A logarithmic distribution in which leading digits of many datasets occur with unequal frequencies. Uniformity of logarithms modulo one explains when leading digits follow this law.
Weyl sum estimate: A bound for exponential sums with polynomial phases, often obtained by differencing. Quantitative versions control how quickly polynomial sequences approach equidistribution.

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Dirichlet's approximation theoremRelated: Diophantine bounds help control how rapidly certain sequences become uniformly distributed.