= Multiple recurrence for circle rotations
Every circle rotation has the <Furstenberg multiple recurrence theorem> property for normalized <Lebesgue measure>. For a rational angle, use a period. For an irrational angle, choose a positive return time with angle close to zero; <translation continuity in L1 on the circle> then makes finitely many translates of a given positive-measure set simultaneously close to that set, and the <union bound> leaves a positive-measure intersection.
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