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The van der Corput lemma for equidistribution

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Quick description

If x_n is a sequence mod 1, such that x_{n+h} - x_n is uniformly distributed modulo 1 for all non-zero h, then x_n is also uniformly distributed modulo 1.

Prerequisites

Undergraduate analysis

Example 1

(Equidistribution of polynomials)

Example 2

(Multiple recurrence in weakly mixing systems)

General discussion

A quantitative version of this fact, known as van der Corput's inequality, is also useful.

(Also mention Bergelson's Hilbert space version.)

Not to be confused with the van der Corput lemma for oscillatory integrals.