The Equidistribution Theorem states that for any irrational number a, the sequence of values a, 2a, 3a, and so on, taken modulo one, is uniformly distributed around the circle formed by wrapping the real line onto itself, meaning the sequence eventually visits every arc of the circle in proportion to that arc's length. It is a special case of the ergodic theorem, obtained by taking the circle's own normalized angle measure as the invariant measure in question.
Facts
Partially Attested
Proof YearSource says the theorem was proved in 1909 and 1910 separately by Weyl, Sierpinski and Bohl; the earlier year is given. StatementFor an irrational number a, the sequence a, 2a, 3a, ... mod 1 is uniformly distributed on the circle R/Z. 1 Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
Proved By
Source Equidistribution theorem (Wikipedia)
Source Equidistribution theorem (Wikipedia)
Sources
1. Equidistribution theorem (Wikipedia)
Intro, sentence 1
is uniformly distributed on the circle
History, sentence 1
this theorem was proved in 1909 and 1910 separately by Hermann Weyl
Proved By: Hermann Weyl, Lead
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.
Proved By: Wacław Sierpiński, Lead
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.
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