# Weyl criterion

From Encyclopedia of Mathematics

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.

#### References

[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: http://encyclopediaofmath.org/index.php?title=Weyl_criterion&oldid=11271

This article was adapted from an original article by B.M. Bredikhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article