The support length of Birkhoff measures for circle rotations is determined by the discrepancy, and the density approximates an isosceles trapezoid for continued-fraction denominators.
Van Aardenne-Ehrenfest, Proof of the Impossibility of a Just Distribution of an Infinite Sequence of Points Over an Interval , Indag
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Birkhoff Measures, Birkhoff Sums, and Discrepancies
The support length of Birkhoff measures for circle rotations is determined by the discrepancy, and the density approximates an isosceles trapezoid for continued-fraction denominators.