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Three-Distance Theorem

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Let be a real number, and consider the arithmetic progression modulo 1. You can think of this as walking along a circle with steps of a fixed length. The three-distance theorem states that the distance between any two consecutive footprints is one of at most three distinct numbers. That is, the circle is partitioned into arcs with at most three distinct lengths.

Contributed by: Eric Rowland (March 2011)
Open content licensed under CC BY-NC-SA

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If is rational, then eventually (namely when is the denominator of ), so the steps are then retraced. In this case there is eventually only one length between distinct consecutive points.

The interesting case is when is irrational. This is the so-called "irrational rotation", and there are either two or three distinct lengths.

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