This is kind of a spin-off of the question asked [here](https://mathoverflow.net/questions/46414/how-quickly-will-billiard-trajectories-cluster). Take the interval $X:=[0,1]$ with $\mu$ being standard Lebesgue measure. Let $f$ be a measure preserving map $f:[0,1]\rightarrow [0,1]$. The Poincaré Recurrence theorem tells us that if I pick a measurable set $E\subset [0,1]$, then under iterations of $f$ almost every point in $E$ returns to $E$ infinitely often, i.e.


$\mu\left({x\in E: \ \exists N, \mbox{ such that } \forall n\geq N, \ f^n(x)\notin E}\right)=0$


Call the set of exceptions above $M$. My question is:



> 
> If I specify an $f$, for what class of sets $E$ in $X$ is $M$ not dense? I am interested in two cases: 
> 
> 1) "dense" with respect to $X$, if $E$ is dense in $X$
> 
> 
> 2) "dense" with respect to $E$
> 
> 
> 


For example, let $E=\mathbb{Q}\cap[0,1]$ and $f(E):=E+\phi$ where $\phi$ is irrational. Then $M=E$, which of course is not a contradiction since $\mu(M)=\mu(E)=0$. On the other hand, throw in the set $H:=\{n\cdot \phi: n\in \mathbb{N}\}$, so that $E':=H\cup E$. In this case we still have $M=E$. As well, it looks like the class of sets I'm interested in is $H\cup {\mbox{not a dense set in X}}$.


Note: My original motivation for asking this was to try and conceptualize the Poincaré recurrence theorem for a human physicist. If I were looking at the phase plot of balls on a billiard table at a specific time, I would only be able to give imprecise measurements of both position and velocity. In this case, it seems that in order to invoke Poincaré recurrence, I would need small intervals around every point to recur perfectly, in the sense that $M=\emptyset$. Perhaps this IS the case if $f$ arises from some nice ODE, but I'm interested in a more general setting. I also don't really want to require that $\mu(M)>0$, which is why I feel asking about denseness is more appropriate. 



Poincaré Recurrence and Dense Sets
If $f$ is minimal, i.e. every orbit is dense, then $E$ is either empty or dense. So it remains to decide if your set is empty. Clearly it is non-empty for positive measure sets. If $\mu(E) = 0$. Then also
$$
 \mu(\bigcup\_{n} f^{-n} E) =0
$$
and for this set the exceptional set is non-empty (and trivially dense).


