Robustness of ergodic dynamical systems

Question

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

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

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.

Answer 1

Yes. Let $\nu$ be a probability measure on $[0,1]$. Let $T$ be the left shift on a sequence space $X:=[0,1]^{\mathbb N}$ equipped with the product $\sigma$-field and the product measure $\mu:=\nu^{\mathbb N}$. Then for every strictly increasing sequence $\{n_k\}$ of positive lower density and $f \in L^1 (X)$ we have $$(*) \quad \lim_{K \to \infty} \frac{1}{K} \sum_{k = 0}^{K-1} f(T^{n_k} (x)) = \int f d\mu \text{ for a.e } x \in X \, ,$$ so $E(T)=0$. To verify $(*)$, it suffices to check it for a dense collection of functions in $L^1$. This reduction follows from a Theorem of Garsia (a version of the Banach principle) as stated e.g. in Theorem 4.2 of [1], see also its application in Theorem 4.3.

If $f$ depends only on the first $q$ coordinates, then $(*)$ follows (for all sequences $n_k$) from the law of large numbers if you separate $n_k$ into $q$ subsequences where each subsequence has gaps at least $q$. Since such functions $f$ are dense, this completes the proof.

[1] Reich, Jakob I. "On the individual ergodic theorem for subsequences." The Annals of Probability 5, no. 6 (1977): 1039-1046. https://projecteuclid.org/euclid.aop/1176995673