4 problems

Ergodic polynomial odd-recurrence conjecture. The set $R$ has nonempty intersection with every infinite arithmetic progression of step size $k$.

Let $f\in L^p_0$ be a specified function. An ergodic invertible measure-preserving transformation is a transformation $T$ preserving the measure and having no nontrivial invariant…

Primality conjecture. In some families of measure-preserving transformations, almost every system is prime.

Naive relative-independence conjecture. The factors $\zeta_0^{T^{\ \!\!\upharpoonright\Gamma_i}}$ are relatively independent over their further factors $\alpha_i$.