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Difference between revisions of "Weyl criterion"

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A fundamental criterion used to solve the problem of the [[Uniform distribution|uniform distribution]] of an infinite sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097700/w0977001.png" /> of arbitrary real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097700/w0977002.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097700/w0977003.png" />, i.e. to establish the existence of the limit
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097700/w0977004.png" /></td> </tr></table>
+
$$
 +
\sum_{n \le N : \alpha \le \{x_n\} \le \beta} \frac{1}{N}
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097700/w0977005.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097700/w0977006.png" /> is the fractional part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097700/w0977007.png" /> (cf. [[Fractional part of a number|Fractional part of a number]]). According to Weyl's criterion, the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097700/w0977008.png" /> is uniformly distributed modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097700/w0977009.png" /> if and only if
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097700/w09770010.png" /></td> </tr></table>
+
$$
 +
\lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N \exp(2\pi i m x_n) = 0
 +
$$
  
for all integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097700/w09770011.png" />. Demonstrated in 1916 by H. Weyl. See [[Weyl method|Weyl method]].
+
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
This article was adapted from an original article by B.M. Bredikhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article