Dye's theorem states that any two ergodic measure-preserving transformations on a standard probability space are orbit equivalent: up to conjugating one of the two, they share the same orbits. Belinskaya's theorem shows that the corresponding cocycles have to behave badly: if they are integrable then the two transformations are flip-conjugate. In a joint work with Carderi, Joseph and Tessera, we show that her result becomes false if one replaces integrability by being in Lp for all p<1. As I will explain, this relies crucially on a new family of Polish groups that we associate to every subadditive function and every measure-preserving transformation.