In a finite measure-preserving system, almost every point of every measurable set returns to infinitely often. The points of that never return form a wandering set: all its inverse images are pairwise disjoint and have equal measure, so that set must be null.
For a probability measure-preserving system, every measurable of positive measure and every integer admit an integer with . The sets are preimages; neither invertibility nor an ergodic transformation is required. The Furstenberg correspondence principle converts this simultaneous return into arithmetic progressions and yields the Szemerédi theorem. The case follows from the Poincare recurrence theorem.
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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The Poincaré recurrence theorem is a fundamental result in the field of dynamical systems and ergodic theory, named after the French mathematician Henri Poincaré. The theorem essentially states that in a closed system where the dynamics are governed by deterministic laws and the system is confined to a finite volume, a system will eventually return to a state very close to its initial conditions after a sufficient amount of time.