I'm solving exercises of the book 'Ergodic throry with a view towards number theory' by my own, and I am too confused about the follwing exercise of the section 2.9, number 3.
Show that the Kakutani-Rokhlin lemma dose not hold for arbitrary sequences of iterates of the map $T$. Specifically, show that for an ergodic measure-preserving system $(X, \mathcal{B},\mu, T)$, sequence $a_1,\cdots,a_n$ of distinct integers, and $\epsilon>0$ it is not always possible to find a measurable set $A$ with the properties that $T^{a_1}(A),…,T^{a_n}(A)$are disjoint and $\mu(\cup^n_{i=1} T^{a_i}(A)>1-\epsilon)$
First I'm confused about whether I can show in general that the set having the required property does not exist or I may suggest only a counterexample.
Second, I suppose I may suggest a counterexample, so for an invertible system we may consider the irrational rotation with an irrational $\alpha$. At first this example seemed useful because we can take an $n$ such that $n\alpha<\epsilon$, so we may get the sets closer to make them intersect, disproving the disjointness of the sets. But I cannot use the fact that the set $A$ has a positive measure, which seems to be related with the regularity of the Lebesgue measure on the circle, but to use open set I have to give up the disjointness condition and to use the compact set I cannot make use of the disjointness condition. How should I proceed here?
Any idea about this problem would be appreciated. I have been looking for all the references mentioned in the book, but never seen any proof.
This is too long for a comment and is not a complete answer. (Also, I think the $\epsilon$ in the question should be replaced with a $1-\epsilon$. This makes more sense).
First, you are supposed to find a counterexample. Namely, find an ergodic system $(X,\mathcal{B},\mu,T)$, distinct integers $a_1,\dots,a_n$, and an $\epsilon > 0$ so that for any $A \in \mathcal{B}$ with $T^{a_1}(A),\dots,T^{a_n}(A)$ pairwise disjoint, we have $\mu(\cup_{i=1}^n T^{a_i}(A)) \le \epsilon$. You couldn't possibly prove that always such an $A$ does not exist since $a_1,\dots,a_n$ could be $1,\dots,n$ in which case there always exists an $A$, by the Kakutani-Rokhlin Lemma.
Second, I like the irrational rotation example. The positivity of the measure of $A$ will help us out via the Lebesgue Density Theorem (this book mentions that theorem a lot). I just tried typing up a proof but ran into some technical issues. I think the idea is roughly as follows. If we take $m \in \mathbb{N}$ so that $\alpha m$ is close to an integer, then for the image under $T^m$ of an interval to be disjoint from itself, the interval needs to be pretty small. And then maybe this will prevent the measure of $A\cup T^m A$ from being more than $1-\epsilon$. The reason I can't make this directly into a proof is that $A$ can be a union of these small intervals - but maybe we would need too many intervals...