The rotation of a circle of circumference is ergodic exactly when is irrational. In the Fourier basis, an invariant function can have a nonzero th coefficient only when .
Every orbit of an irrational rotation of the circle visits a half-open interval with limiting frequency equal to its length. One proof sandwiches the orbit of an arbitrary point between inner and outer intervals along a nearby orbit for which the Birkhoff ergodic theorem holds.
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Irrational rotation refers to a concept in mathematics, particularly in the field of dynamical systems and geometry. It typically involves rotations by an angle that is an irrational multiple of \( \pi \), meaning that the rotation angle cannot be expressed as a simple fraction. ### Key aspects of irrational rotation: 1. **Continuous Rotation**: When an object (like a point on a circle or a plane) is rotated continuously by an irrational angle, it never returns to its original position.