The block $617$ occurs infinitely many times in the decimal expansion of almost every $x ∈ [0,1]$

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.

• There's something fishy about the first claim. The number all of whose decimal digits are (say) 3, except for the initial block of 617, viz $x=.61733333...$ looks like a counter example. Maybe you mis-stated what you meant to ask? – kimchi lover Mar 13 at 21:43
• The first claim as stated is plainly wrong. I wonder if what is meant is that it's true of almost every $x \in [0, 1]$. – Robert Shore Mar 13 at 21:51
• The correct is for almost everywhere number $x∈[0,1]$ – Ricardo Freire Mar 13 at 22:03
• Borel-Cantelli? – Jair Taylor Mar 13 at 22:16
• I can not see relation in using Borel cantelli and explain the existence of the interval $J'$ – Ricardo Freire Mar 13 at 22:22

$$\ J' = \left[\frac{10^3j+617}{10^{k+3}}, \frac{10^3j+618}{10^{k+3}} \right)$$