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.
Schoissengeier, On the Discrepancy of (n α II), Journal of Number Theory,24, 54-64 ( 1986)
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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.