Equidistribution theorem

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.

In modern formulations, it is asked under what conditions the identity might hold, given some general sequence bk.

Similarly, for the sequence bk = 2ka, for every irrational a, and almost all x, there exists a function ƒ for which the sum diverges.

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.

Illustration of filling the unit interval (horizontal axis) with the first n terms using the equidistribution theorem with four common irrational numbers, for n from 0 to 999 (vertical axis). The 113 distinct bands for π are due to the closeness of its value to the rational number 355/113. Similarly, the 7 distinct groups are due to π being approximately 22/7.
(click for detailed view)