Weyl criterion

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A fundamental criterion used to solve the problem of the uniform distribution of an infinite sequence of arbitrary real numbers modulo , i.e. to establish the existence of the limit

where and is the fractional part of (cf. Fractional part of a number). According to Weyl's criterion, the sequence is uniformly distributed modulo if and only if

for all integers . Demonstrated in 1916 by H. Weyl. See Weyl method.


[1] J.W.S. Cassels, "An introduction to diophantine approximation" , Cambridge Univ. Press (1957)
How to Cite This Entry:
Weyl criterion. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by B.M. Bredikhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article