Prove that, for almost everywhere number $x ∈ [0,1]$ whose decimal expansion contains the
block $617$ (for instance, $x = 0.3375617264 ···$), that block occurs infinitely
many times in the decimal expansion of $x$. Even more, the block $617$ occurs
infinitely many times in the decimal expansion of almost every $x ∈ [0,1]$.
The first part I did taking $E = [0, 617, 0, 618)$ which has positive measure and use that the Lebesgue measure is $f$-invariant where $f(x)=10x- \lfloor 10x \rfloor$ and apply the recurrence theorem
of Poincaré.
The second part the book gives a hint:
Note that every interval $J = [ j / 10^k, (j + 1) / 10^k)$ contains subinterval $J'$ such that
$\frac{m (J')}{m (J)} = \frac{1}{10^{3}}$ and $f^k (x) ∈ E$ for all $x ∈ J'$.
Using Theorem $A.2.14$, conclude that
all $x ∈ [0, 1]$ has at least one iterate in $E$. Now the second statement in the exercise the is consequence of the first part.
Theorem $A.2.14.$ Let A be a measurable subset of $R^d$ with Lebesgue measure
$m(A)$ positive. Then $m$-almost every $a ∈ A$ is a density point of $A$.
I can not see that there is this interval $ J '$. Someone can explain.
Best Answer
$$\ J' = \left[\frac{10^3j+617}{10^{k+3}}, \frac{10^3j+618}{10^{k+3}} \right) $$