In mathematics, the equidistribution theorem is the statement that the sequence Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ...

a, 2a, 3a, ... mod 1

is uniformly distributed on the unit interval, when a is an irrational number. It is a special case of the ergodic theorem. While this theorem was proved in 1909 and 1910 separately by Hermann Weyl, Wacław Sierpiński and P. Bohl, variants of this theorem continue to be studied to this day. In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ... In mathematics, an irrational number is any real number that is not a rational number, i. ... In mathematics, a measure-preserving transformation T on a probability space is said to be ergodic if the only measurable sets invariant under T have measure 0 or 1. ... 1909 (MCMIX) was a common year starting on Friday (see link for calendar). ... -1... Hermann Weyl Hermann Weyl (November 9, 1885 - December 8, 1955) was a German mathematician. ... WacÅ‚aw Franciszek SierpiÅ„ski (March 14, 1882 — October 21, 1969), a Polish mathematician, was born and died in Warsaw. ...

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 a is the nth prime. Vinogradov's proof was a byproduct of the odd Goldbach conjecture, that every large odd number is the sum of three primes. 1916 (MCMXVI) is a leap year starting on Saturday (link will take you to calendar) // Events January-February January 1 -The first successful blood transfusion using blood that had been stored and cooled. ... 1935 (MCMXXXV) was a common year starting on Tuesday (link will take you to calendar). ... Ivan Matveevich Vinogradov (September 14, 1891–March 20, 1983) was a Russian mathematician, who was one of the creators of modern analytic number theory, and also the dominant figure in mathematics in the USSR. He was born in the Velikiye Luki district, Pskov Oblast. ... In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ...

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. George David Birkhoff (21 March 1884 - 12 November 1944) was an American mathematician, and one of the most important leaders in mathematics in the USA in his generation. ... 1931 (MCMXXXI) is a common year starting on Thursday. ... 1933 (MCMXXXIII) was a common year starting on Sunday (link will take you to calendar). ... In mathematics, the phrase almost all has a number of specialised uses. ... In mathematics, the Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. ... Jean Bourgain (born February 28, 1954, Ostende, Belgium), is a professor of mathematics at the Institute for Advanced Study. ... 1988 (MCMLXXXVIII) was a leap year starting on a Friday of the Gregorian calendar. ...

Specifically, Khinchin showed that the identity

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

might hold, given some general sequence b_k. This is a page about mathematics. ...

One noteworthy result is that the sequence 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. For the case of multiples of a, Weyl's criterion reduces the problem to summing finite geometric series. In mathematics, an exponential sum may be a finite Fourier series (i. ... In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ...

See also

• Diophantine approximation
• Low-discrepancy sequence

In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers. ... In mathematics, a low-discrepancy sequence is a sequence with the property that for all N, the subsequence x1, ..., xN is almost uniformly distributed (in a sense to be made precise), and x1, ..., xN+1 is almost uniformly distributed as well. ...

Historical references

• P. 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'sce Ershcheinung und verwandte Konvergenzphänomene, (1910) Rendiconti del Circolo Matematico di Palermo, 330, pp377-407.
• W. Sierpinski, Sur la valeur asymptotique d'une certaine sommme, (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. Focuses on methods developed by Bourgain.)