Rosenblatt–Wierdl conjecture on divergence for zero-density ergodic averages

Let $a_n$ be a sequence of integers with zero upper Banach density, meaning

$$\limsup_{|I| \to \infty \text{ an interval}} \frac{|I \cap \{a_n\}|}{|I|}=0.$$

For any probability space $(X,\mu)$ equipped with an aperiodic, measure-preserving transformation $T:X\to X$, there exists $f\in L^1(X)$ such that

$$\frac{1}{N}\sum_{n\leq N}T^{a_n}f, \qquad T^k f(x):=f(T^k x),$$

Rosenblatt–Wierdl conjecture. does not converge almost everywhere. This conjecture concerns divergence of sparse ergodic averages at the $L^1$ endpoint; the paper studies quantitative convergence for particular deterministic and random sparse sequences, while the general assertion remains unresolved.