Let say that an infinite subsets $A$ of $\mathbb{N}$ is "nice w.r.to ergodic limits", if it can replace $\mathbb{N}$ in the individual ergodic theorem, that is, if it is such that the following statement is true:

For any probability space $(X,\Sigma,\mu),$ for any measure-preserving transformation $T$ on $X,$ for any $f\in L^1(X,\mu)$ the ergodic means along $A,$

$$M(f,T,A,t,x):=|\{j\in A\, : j\leq t\}|^{-1}\sum_{j\in A,\, j\leq t} > f(T^{j}x)$$

converge for a.e. $x\in X$ to the conditional expectation w.r.to the $T$ invariant $\sigma$-algebra, $\mathbb{E}(f|\Sigma_T),$ as $t \to +\infty.$

So $\mathbb{N}$ itself is nice in this sense, by Birkhoff's theorem; if $A$ is nice and $m$ is a positive integer, the set of translates $A+m$ is nice (the set of convergence with $T$ along $A+m$ coinciding with the $T^{\, m}$ pre-image of the set of convergence with $T$ along $A$). Also, a disjoint union of two nice sets is nice.

Is there any other structure on the family of these sets? What about e.g. the union of two of them? (at a glance it seems to me that something more can be said for the analogous cases of other ergodic theorems, e.g. for the $L^p$ convergence.

Looking at this very related question, and its answer, make me think that the situation may be non-trivial and interesting enough, so that it should have been studied.