Difference between revisions of "Weyl criterion"
From Encyclopedia of Mathematics
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| − | A fundamental criterion used to solve the problem of the [[Uniform distribution|uniform distribution]] of an infinite sequence | + | A fundamental criterion used to solve the problem of the [[Uniform distribution|uniform distribution]] of an infinite sequence $(x_n)$ of arbitrary real numbers $x_n$ modulo 1, i.e. to establish that the limit as $N \rightarrow \infty$ of |
| − | + | $$ | |
| + | \sum_{n \le N : \alpha \le \{x_n\} \le \beta} \frac{1}{N} | ||
| + | $$ | ||
| − | where | + | exists and is equal to $\beta - \alpha$, where $ 0 \le \alpha \le \beta \le 1 $ and $\{x_n\}$ is the fractional part of $x_n$ (cf. [[Fractional part of a number|Fractional part of a number]]). Weyl's criterion states that the sequence $(x_n)$ is uniformly distributed modulo 1 if and only if |
| − | + | $$ | |
| + | \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N \exp(2\pi i m x_n) = 0 | ||
| + | $$ | ||
| − | for all integers | + | for all integers $m \ne 0$. Demonstrated in 1916 by H. Weyl. See [[Weyl method|Weyl method]]. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.W.S. Cassels, "An introduction to diophantine approximation" , Cambridge Univ. Press (1957)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.W.S. Cassels, "An introduction to diophantine approximation" , Cambridge Univ. Press (1957)</TD></TR></table> | ||
Revision as of 17:55, 18 August 2013
A fundamental criterion used to solve the problem of the uniform distribution of an infinite sequence $(x_n)$ of arbitrary real numbers $x_n$ modulo 1, i.e. to establish that the limit as $N \rightarrow \infty$ of
$$ \sum_{n \le N : \alpha \le \{x_n\} \le \beta} \frac{1}{N} $$
exists and is equal to $\beta - \alpha$, where $ 0 \le \alpha \le \beta \le 1 $ and $\{x_n\}$ is the fractional part of $x_n$ (cf. Fractional part of a number). Weyl's criterion states that the sequence $(x_n)$ is uniformly distributed modulo 1 if and only if
$$ \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N \exp(2\pi i m x_n) = 0 $$
for all integers $m \ne 0$. 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
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