Let $\mathbf X := (X, \mathcal F, \mu)$ be a standard probability space. For an ergodic measure preserving transformation $T$, we define the ergodic robustness $\mathcal R(T)$ of $T$ as follows:

For $0 \leq r \leq 1$, let $C_r \subset \mathbb N^{\mathbb N}$ be the subset of monotonically increasing sequences whose natural density exists and is greater than of equal to $r$.

Define the quantity $E(T)$ by

$$E(T) := \inf \{r \in [0, 1]| \text{ For all } f \in L^1 (X), n_k \in C_r, \lim_{K \to \infty} \frac{1}{K} \sum_{k = 0}^{K-1} f(T^{n_k} (x)) = \int f d\mu \text{ for a.e } x \in X\}.$$

Finally, define $\mathcal R(T) = 1 - E(T)$.

Question: Do there exist ergodic measure preserving transformations $T$ with $\mathcal R(T)$ arbitrarily close to $1$? That is, for every $\varepsilon > 0$, does there exist an ergodic transformation $T$ with $\mathcal R(T) > 1 - \varepsilon$?

Remark: The choice to use $\mathcal R(T)$ instead of $E(T)$ is purely aesthetic to fit the terminology, and is not very important.