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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 $(x_n)$ of arbitrary real numbers $x_n$ modulo 1, i.e. to establish that the limit as $N \rightarrow \infty$ of
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A fundamental criterion used to solve the problem of the uniform [[distribution modulo one]] 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
  
 
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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
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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]]).  Weyl's criterion states that the sequence $(x_n)$ is uniformly distributed modulo 1 if and only if
  
 
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for all integers $m \ne 0$.  Demonstrated in 1916 by H. Weyl. See [[Weyl method|Weyl method]].
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for all integers $m \ne 0$.  Demonstrated in 1916 by H. Weyl. See [[Weyl method]].
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====References====
 
====References====
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<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>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  J.W.S. Cassels,  "An introduction to diophantine approximation" , Cambridge University Press (1957)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top"> H. Weyl, "Ueber die Gleichverteilung von Zahlen mod. Eins,". ''Math. Ann.'' '''77''', no.3 (1916) 313–352. {{DOI|10.1007/BF01475864}}</TD></TR>
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</table>

Latest revision as of 22:16, 12 March 2018


A fundamental criterion used to solve the problem of the uniform distribution modulo one 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 University Press (1957)
[2] H. Weyl, "Ueber die Gleichverteilung von Zahlen mod. Eins,". Math. Ann. 77, no.3 (1916) 313–352. DOI 10.1007/BF01475864
How to Cite This Entry:
Weyl criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weyl_criterion&oldid=30171
This article was adapted from an original article by B.M. Bredikhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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