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In [[mathematics]], in the theory of [[Diophantine approximation]], '''Weyl's criterion''' states that a [[sequence]] <math>(x_{n})</math> of [[real number]]s is [[equidistributed mod 1]] if and only if for all non-zero integers <math>\ell</math> | |||
:<math>\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{j=0}^{n-1}e^{2\pi i \ell x_{j}}=0.</math> | |||
The criterion is named after, and was first formulated by, [[Hermann Weyl]].<ref>{{cite journal|last=Weyl|first=H.|title=Ueber die Gleichverteilung von Zahlen mod. Eins,|journal=Math. Ann.|year=1916|volume=77|pages=313–352 | doi = 10.1007/BF01475864 | issue = 3}}</ref> It allows to equidistribution questions to bounds on [[exponential sum]]s, a fundamental and general method. | |||
The criterion extends naturally to higher [[dimension]]s. A sequence | |||
:<math>x_{n}\in\mathbb{R}^{k}</math> | |||
is [[equidistributed mod 1]] if and only if for any <math>\ell\in\mathbb{Z}^{k}</math> which is not the zero vector, | |||
:<math>\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{j=0}^{n-1}e^{2\pi i (\ell \cdot x_{j})}=0.</math> | |||
==See also== | |||
A quantitative form of the Weyl criterion is given by the [[Erdős–Turán inequality]]. | |||
==References== | |||
<references /> | |||
==External links== | |||
* [http://mathworld.wolfram.com/WeylsCriterion.html Weyl's Criterion at Mathworld] | |||
* [http://planetmath.org/encyclopedia/WeylsCriterion.html Weyl's Criterion at Planetmath] | |||
* [http://www.maths.manchester.ac.uk/~cwalkden/ergodic-theory/lecture06.pdf Lecture notes with proof] | |||
[[Category:Diophantine approximation]] | |||
[[Category:Dynamical systems]] | |||
[[Category:Ergodic theory]] | |||
Revision as of 23:50, 23 January 2014
In mathematics, in the theory of Diophantine approximation, Weyl's criterion states that a sequence of real numbers is equidistributed mod 1 if and only if for all non-zero integers
The criterion is named after, and was first formulated by, Hermann Weyl.[1] It allows to equidistribution questions to bounds on exponential sums, a fundamental and general method.
The criterion extends naturally to higher dimensions. A sequence
is equidistributed mod 1 if and only if for any which is not the zero vector,
See also
A quantitative form of the Weyl criterion is given by the Erdős–Turán inequality.
References
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