Belinskaya’s theorem is optimal

Belinskaya's theorem states that given an ergodic measure-preserving transformation, any other transformation with the same orbits and $\mathrm{L}^1$ cocycle must be flip-conjugate to it. Our main result shows this is optimal: for all $p<1$ integrability condition on cannot relaxed being in $\mathrm{L}^p$. This also allows us answer a question of Kerr Li: transformations, Shannon orbit equivalence doesn't boil down flip-conjugacy.