two continuous functions on the interval, whose integrals differ by an arbitrary ε. By an argument similar to the proof of the Riemann integral criterion, it is possible to extend the result to any interval indicator function f, thereby proving equidistribution modulo 1 of the given sequence. ∎ |} Generalizations A quantitative form of Weyl's criterion is given by the Erdős–Turán inequality. Weyl's criterion extends naturally to higher dimensions, assuming the natural generalization of the definition of equidistribution modulo 1: The sequence vn of vectors in Rk is equidistributed modulo 1 if and only if for any non-zero vector ℓ ∈ Zk, Example of usage Weyl's criterion can be used to easily prove the equidistribution theorem, stating that the sequence of multiples 0, α, 2α, 3α, ... of some real number α is equidistributed modulo 1 if and only if α is irrational. Suppose α is irrational and denote our sequence by aj = jα (where j starts from 0, to simplify the formula later). Let ℓ ≠ 0 be an integer. Since α is irrational, ℓα can never be an integer, so can never be 1. Using the formula for the sum of a finite geometric series, a finite bound that does not depend on n. Therefore, after dividing by n and letting n tend to infinity, the left hand side tends to zero, and Weyl's criterion is satisfied. Conversely, notice that if α is rational then this sequence is not equidistributed modulo 1, because there are only a finite number of options for the fractional part of aj = jα. Complete uniform distribution A sequence of real numbers is said to be k-uniformly distributed mod 1 if not only the sequence of fractional parts is uniformly distributed in but also the sequence , where is defined as , is uniformly distributed in . A sequence of real numbers is said to be completely uniformly distributed mod 1 it is -uniformly distributed for each natural number . For example, the sequence is uniformly distributed mod 1 (or 1-uniformly distributed) for any irrational number , but is never even 2-uniformly distributed. In contrast, the sequence is completely uniformly distributed for almost all (i.e., for all except for a set of measure 0). van der Corput's difference theorem A theorem of Johannes van der Corput states that if for each h the sequence sn+h − sn is uniformly distributed modulo 1,