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| In [[mathematics]], the '''equidistribution theorem''' is the statement that the sequence
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| :''a'', 2''a'', 3''a'', ... mod 1 | |
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| is [[Equidistributed sequence|uniformly distributed]] on the [[circle]] <math>\mathbb{R}/\mathbb{Z}</math>, when ''a'' is an [[irrational number]]. It is a special case of the [[ergodic theorem]] where one takes the normalized angle measure <math>\mu=\frac{d\theta}{2\pi}</math>.
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| ==History==
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| While this theorem was proved in 1909 and 1910 separately by [[Hermann Weyl]], [[Wacław Sierpiński]] and [[Piers Bohl]], variants of this theorem continue to be studied to this day.
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| In 1916, Weyl proved that the sequence ''a'', 2<sup>2</sup>''a'', 3<sup>2</sup>''a'', ... mod 1 is uniformly distributed on the unit interval. In 1935, [[Ivan Vinogradov]] proved that the sequence ''p''<sub>''n''</sub> ''a'' mod 1 is uniformly distributed, where ''p''<sub>''n''</sub> is the ''n''th [[prime number|prime]]. Vinogradov's proof was a byproduct of the [[odd Goldbach conjecture]], that every sufficiently large odd number is the sum of three primes.
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| [[George Birkhoff]], in 1931, and [[Aleksandr Khinchin]], in 1933, proved that the generalization ''x'' + ''na'', for [[almost all]] ''x'', is equidistributed on any [[Lebesgue measurable]] subset of the unit interval. The corresponding generalizations for the Weyl and Vinogradov results were proven by [[Jean Bourgain]] in 1988.
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| Specifically, Khinchin showed that the identity
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| :<math>\lim_{n\to\infty} \frac{1}{n} \sum_{k=1}^n | |
| f( (x+ka) \mod 1 ) = \int_0^1 f(y)\,dy </math>
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| holds for almost all ''x'' and any Lebesgue integrable function ƒ. In modern formulations, it is asked under what conditions the identity
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| :<math>\lim_{n\to\infty} \frac{1}{n} \sum_{k=1}^n
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| f( (x+b_ka) \mod 1 ) = \int_0^1 f(y)\,dy </math>
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| might hold, given some general [[sequence]] ''b''<sub>''k''</sub>.
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| One noteworthy result is that the sequence 2<sup>''k''</sup>''a'' mod 1 is uniformly distributed for almost all, but not all, irrational ''a''. Similarly, for the sequence ''b''<sub>''k''</sub> = 2<sup> ''k''</sup>, for every irrational ''a'', and almost all ''x'', there exists a function ƒ for which the sum diverges. In this sense, this sequence is considered to be a '''universally bad averaging sequence''', as opposed to ''b''<sub>''k''</sub> = ''k'', which is termed a '''universally good averaging sequence''', because it does not have the latter shortcoming.
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| A powerful general result is [[Weyl's criterion]], which shows that equidistribution is equivalent to having a non-trivial estimate for the [[exponential sum]]s formed with the sequence as exponents. For the case of multiples of ''a'', Weyl's criterion reduces the problem to summing finite [[geometric series]].
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| ==See also==
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| * [[Diophantine approximation]]
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| * [[Low-discrepancy sequence]]
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| ==References==
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| ===Historical references===
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| * P. Bohl, (1909) ''Über ein in der Theorie der säkutaren Störungen vorkommendes Problem'', ''J. reine angew. Math.'' '''135''', pp, 189–283.
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| * {{cite journal | last1 = Weyl | first1 = H. | year = 1910 | title = Über die Gibbs'sche Erscheinung und verwandte Konvergenzphänomene | url = | journal = [[Rendiconti del Circolo Matematico di Palermo]] | volume = 330 | issue = | pages = 377–407 }}
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| * W. Sierpinski, (1910) ''Sur la valeur asymptotique d'une certaine somme'', ''Bull Intl. Acad. Polonmaise des Sci. et des Lettres'' (Cracovie) '''series A''', pp. 9–11.
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| * {{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 |postscript = .}}
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| * {{cite journal | first1= G. D. |last1=Birkhoff |url=http://www.pnas.org/cgi/reprint/17/12/656 |title = Proof of the ergodic theorem |year=1931 |volume = 17 |issue = 12 |pages= 656–660 | pmc = 1076138 |postscript = . |pmid=16577406 |journal=Proc. Natl. Acad. Sci. U.S.A. |doi=10.1073/pnas.17.12.656}}
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| * {{cite journal | first1= A. |last1 = Ya. Khinchin |title = Zur Birkhoff's Lösung des Ergodensproblems |year= 1933 | journal = Math. Ann. | volume = 107| pages = 485–488 |postscript = . | doi = 10.1007/BF01448905}}
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| ===Modern references===
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| * Joseph M. Rosenblatt and Máté Weirdl, ''Pointwise ergodic theorems via harmonic analysis'', (1993) appearing in ''Ergodic Theory and its Connections with Harmonic Analysis, Proceedings of the 1993 Alexandria Conference'', (1995) Karl E. Petersen and Ibrahim A. Salama, ''eds.'', Cambridge University Press, Cambridge, ISBN 0-521-45999-0. ''(An extensive survey of the ergodic properties of generalizations of the equidistribution theorem of [[shift map]]s on the [[unit interval]]. Focuses on methods developed by Bourgain.)''
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| * Elias M. Stein and Rami Shakarchi, ''Fourier Analysis. An Introduction'', (2003) Princeton University Press, pp 105–113 ''(Proof of the Weyl's theorem based on Fourier Analysis)''
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| [[Category:Ergodic theory]]
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| [[Category:Diophantine approximation]]
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| [[Category:Theorems in number theory]]
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