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.
Kesten, On a conjecture of Erd¨ os and Sz¨ usz related to uniform distribution mod 1 , Acta Arith
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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.