Equidistribution theorem

In mathematics, the equidistribution theorem is the statement that the sequence

a, 2a, 3a, . Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and . . mod 1

is uniformly distributed on the unit interval, when a is an irrational number. In Mathematics, a bounded sequence { s 1 s 2 s 3 …} of Real numbers is said to be equidistributed In Mathematics, the unit interval is the interval, that is the set of all Real numbers x such that zero is less than or equal to x In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction It is a special case of the ergodic theorem. Ergodic theory is a branch of Mathematics that studies Dynamical systems with an Invariant measure and related problems

1 History
2 See also
3 References
- 3.1 Historical references
- 3.2 Modern references

History

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. Hermann Klaus Hugo Weyl ( 9 November 1885 – 8 December 1955) was a German Mathematician. Wacław Franciszek Sierpiński ( March 14 1882 — October 21 1969) (ˈvaʦwaf fraɲˈʨiʂɛk ɕɛrˈpʲiɲskʲi a Polish Mathematician Piers Bohl ( October 23, 1865 - December 25, 1921) was a Latvian Mathematician

In 1916, Weyl proved that the sequence a, 2²a, 3²a, . . . mod 1 is uniformly distributed on the unit interval. In 1935, Ivan Vinogradov proved that the sequence p_n a mod 1 is uniformly distributed, where p_n is the nth prime. Ivan Matveevich Vinogradov (Иван Матвеевич Виноградов September 14, 1891 &ndash March 20, 1983) was a Russian In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 Vinogradov's proof was a byproduct of the odd Goldbach conjecture, that every large odd number is the sum of three primes. In Number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem

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. George David Birkhoff ( 21 March 1884, Overisel Michigan - 12 November 1944, Cambridge Massachusetts) was an American Aleksandr Yakovlevich Khinchin ( Russian Алекса́ндр Я́ковлевич Хи́нчин French Alexandre Khintchine ( July 19, 1894 See also Generic property In Mathematics, the phrase almost all has a number of specialised uses In Mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a Length, Area or Volume to The corresponding generalizations for the Weyl and Vinogradov results were proven by Jean Bourgain in 1988. Jean Bourgain (b February 28, 1954, Ostend) is a Belgian Mathematician, noted as a prolific problem-solver

Specifically, Khinchin showed that the identity

$\lim_{n\to\infty} \frac{1}{n} \sum_{k=1}^n f( (x+ka) \mod 1 ) = \int_0^1 f(y)\,dy$

holds for almost all x and any Lebesgue integrable function f. In modern formulations, it is asked under what conditions the identity

$\lim_{n\to\infty} \frac{1}{n} \sum_{k=1}^n f( (x+b_ka) \mod 1 ) = \int_0^1 f(y)\,dy$

might hold, given some general sequence $b k$ . In Mathematics, a sequence is an ordered list of objects (or events

One noteworthy result is that the sequence $2 k a$ mod 1 is uniformly distributed for almost all, but not all, irrational a. Similarly, for the sequence $b k = 2 k$ , for every irrational a, and almost all x, there exists a function f for which the sum diverges. In this sense, this sequence is considered to be a universally bad averaging sequence, as opposed to $b k = k$ , which is termed a universally good averaging sequence, because it does not have the latter shortcoming.

A powerful general result is Weyl's criterion, which shows that equidistribution is equivalent to having a non-trivial estimate for the exponential sums formed with the sequence as exponents. In Mathematics, in the theory of Diophantine approximation, Weyl's criterion states that a Sequence (x_{n} of Real numbers is In Mathematics, an exponential sum may be a finite Fourier series (i For the case of multiples of a, Weyl's criterion reduces the problem to summing finite geometric series. In Mathematics, a geometric series is a series with a constant ratio between successive terms.

References

Historical references

P. In Number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of Real numbers by Rational In Mathematics, a low-discrepancy sequence is a Sequence with the property that for all values of N, its subsequence x 1. Bohl, Über ein in der Theorie der säkutaren Störungen vorkommendes Problem, (1909), J. reine angew. Math. 135, pp, 189-283.
H. Weyl, Über die Gibbs'sche Erscheinung und verwandte Konvergenzphänomene, (1910) Rendiconti del Circolo Matematico di Palermo, 330, pp. 377-407.
W. Sierpinski, Sur la valeur asymptotique d'une certaine somme, (1910), Bull Intl. Acad. Polonmaise des Sci. et des Lettres (Cracovie) series A, pp. 9-11.
H. Weyl, Über die Gleichverteilung von Zählen mod. Eins, (1916) Math. Ann. 77, pp. 313-352.
G. D. Birkhoff, Proof of the ergodic theorem, (1931), Proceedings of the National Academy of Sciences USA, 17, pp. 656-660.
A. Ya. Khinchin, Zur Birkhoff's Lösung des Ergodensproblems, (1933), Math. Ann. 107, pp. 485-488.

Modern references

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 maps on the unit interval. In Mathematics, and in particular Functional analysis, the shift operators are examples of Linear operators important for their simplicity and natural occurrence In Mathematics, the unit interval is the interval, that is the set of all Real numbers x such that zero is less than or equal to x Focuses on methods developed by Bourgain. )
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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Contents

History

See also

References

Historical references

Modern references