For almost every point in a positive-measure set under an ergodic flow, there are arbitrarily late return times to the set with exactly zero integral of a zero-mean function.
Moshchevitin, Recurrence of the integral of a smoot h three-frequency conditionally periodic function, Math
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Around Krygin-Atkinson, Shneiberg theorems, the recurrence with zero integrals
For almost every point in a positive-measure set under an ergodic flow, there are arbitrarily late return times to the set with exactly zero integral of a zero-mean function.