TITLE: Historical behavior of the Birkhoff averages
QUESTION [5 upvotes]: Birkoff  Ergodic Theorem: Let  be $(X,\mathcal{A},\mu)$ a probability space, $T:X\to X$ a measurable tranformation preserving $\mu$ and $f\in L^{1}(\mu)$, then there exists a 
$\Sigma \subset X$ and $\mu(\Sigma)=1$ such that  the limit
$$
\widetilde{f}(x)=\lim_{n\to\infty}\dfrac{1}{n}\sum_{j=0}^{n-1}f(T^j(x))
$$
exists  for all $x\in \Sigma.$ Also   $~\widetilde{f}\circ T= \widetilde{f} $     ,$\widetilde{f}\in L^1(\mu)$ and $\int fd\mu=\int \widetilde{f}d\mu.$
My problem: Let be for example $T(x)=2x~\text{mod} 1$ the doubling map. I'm looking for a concrete example of a Borelian and point $x$ such that  the limit 
$$
\widetilde{1}_A(x)=\lim_{n\to\infty}\dfrac{1}{n}\sum_{j=0}^{n-1}1_A(T^j(x))
$$
not exists.
Obs.: The idea for this is more or less clear to me, It comes to choosing a $ x $ that is not a preperiodic point, than  the borelian searched is 
$$ 
A=\{   T^nx, ~n\in I \}
$$
where $ I $ is a set of type $I=\{1,2, ,\ldots,a_1, \underbrace{a_2,a_2+1,a_2+2\ldots}_{a_2=2^{a_1}}, \underbrace{a_3,a_3+1,a_3+2\ldots}_{a_3=2^{a_2}}, a_4,\ldots \}$
hence, the  of Birkhoff averages of $x$  will be oscillating between 0 and 1.
My problem is that I'm not able to show this with a concrete example, with all Epsilons and deltas

REPLY [3 votes]: Take $A = [1/2,1]$ and 
$$x = \sum_{k=0}^\infty \sum_{j=3^{2k+1}}^{3^{2k+2}-1} 2^{-j} $$
i.e. the base-2 "decimal" expansion of $x$ has $0$ in position $j$ if
$\lfloor \log_3(j) \rfloor$ is even and $1$ if it is odd.
Note that $\frac{1}{n} \sum_{j=0}^{n-1} 1_A(T^j x)$ is the fraction of
$1$'s in positions $1$ to $n$ of the expansion of $x$, which is at least 
$2/3$ if $n = 3^{2k}$ and at most $1/3$ if $n = 3^{2k+1}$.