Equidistribution theorem

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

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

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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, Wac?aw Sierpi?ski and P. Bohl, variants of this theorem continue to be studied to this day.

Related Topics:
Uniformly distributed - Unit interval - Irrational number - Ergodic theorem - 1909 - 1910 - Hermann Weyl - Wac?aw Sierpi?ski - P. Bohl

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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 pn a mod 1 is uniformly distributed, where pn a is the n-th prime. Vinogradov's proof was a byproduct of the odd Goldbach conjecture, that every large odd number is the sum of three primes.

Related Topics:
1916 - 1935 - Ivan Vinogradov - Odd Goldbach conjecture

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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.

Related Topics:
George Birkhoff - 1931 - Aleksandr Khinchin - 1933 - Almost all - Lebesgue measurable - Jean Bourgain - 1988

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Specifically, Khinchin showed that the identity

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:lim_{n oinfty} sum_{k=1}^n

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f( (x+ka) mod 1 ) = int_0^1 f(y)dy

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

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:lim_{n oinfty} sum_{k=1}^n

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f( (x+b_ka) mod 1 ) = int_0^1 f(y)dy

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might hold, given some general sequence b_k.

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One noteworthy result is that the sequence 2^ka 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.

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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 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.

Related Topics:
Weyl's criterion - Exponential sum - Geometric series

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