Equidistribution theorem

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

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

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.

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_na mod 1 is uniformly distributed, where p_n is the nth prime. Vinogradov's proof was a byproduct of the odd Goldbach conjecture, that every sufficiently large odd number is the sum of three primes.

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