I am trying to solve both 1.3.9 and 1.3.10 from the book "foundation of Ergodic Theory" by Viana - Oliveira. I have solved 1.3.9, but I am totally stuck on 1.3.10. I posted my attempt below.
Here is the text:
My attempt so far:
Since it's the first chapter of the book I should be able so solve the problem by only using topology, Poincaré Recurrency Theorem and Liouville Theorem. I recall that $\theta_1, \ldots, \theta_d$ are "rationally independent" if whenever we have $n_0, n_1, \ldots, n_d$ such that: $$ n_0 + \sum_{i = 1}^d n_i d_i = 0 \implies n_0 = n_1 = \ldots = n_d = 0.$$
I Tried by contradiction: suppose That there is a non empty subset of $\mathbb{T}^d$ that is invariant under $R_{\theta}$ and that is not dense. There there is $U$ open such that $V \cap U = \emptyset$. That's where I am stuck. I tried to study the iterations $R_{\theta}^n(U)$ in order to find a countable collection of disjoint sets to get a contradiction(if we find such collection the Torus would have infinite measure). But I can't see how to use the hypothesis on $\theta$ to show it.
EDIT: I maybe made some progress but still blind. So we know that $V$ is $R_{\theta}$ invariant and that $U \cap V = \emptyset$. That means that $\forall n \in \mathbb{Z}$ We would have $R^{n}_{\theta}(U) \cap V = \emptyset$, which implies that $R^n_{\theta}(U) \subset \mathbb{T}^n-V$. Since they have all positive measure they must intersect somewhere so there are $h, l \in \mathbb{Z}$ such that $R^{h}_{\theta}(U) \cap R^{l}_{\theta}(U) \neq \emptyset$. Which means there is $x \in U$ and $y \in R^{h-l}(U)$ such that $y = R^{h-l}(x)$. Now by rational independence of $\theta$ we can't have taht $y, x \in \mathbb{Q}^n \cap \mathbb{T}^n$. This is where I am stuck now. I tried to study the orbits of rational points in $U$, indeed if we pick $p \in \mathbb{Q}^n \cap U$ then we can't have that the orbit repeats itself(i.e. $R^k(p) = R^{k'}(p)$ for some integers) and the orbits of rational points can't intersect(other whise rational independence is violeted). But I don't know how to put all this together. Any hint is appreciated.