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

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

is uniformly distributed on the circle $\mathbb {R} /\mathbb {Z}$ , when a is an irrational number. It is a special case of the ergodic theorem where one takes the normalized angle measure $\mu ={\frac {d\theta }{2\pi }}$ .