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.

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    $\begingroup$ 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? $\endgroup$ – kimchi lover Mar 13 at 21:43
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    $\begingroup$ The first claim as stated is plainly wrong. I wonder if what is meant is that it's true of almost every $x \in [0, 1]$. $\endgroup$ – Robert Shore Mar 13 at 21:51
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    $\begingroup$ The correct is for almost everywhere number $x∈[0,1]$ $\endgroup$ – Ricardo Freire Mar 13 at 22:03
  • $\begingroup$ Borel-Cantelli? $\endgroup$ – Jair Taylor Mar 13 at 22:16
  • $\begingroup$ I can not see relation in using Borel cantelli and explain the existence of the interval $J'$ $\endgroup$ – Ricardo Freire Mar 13 at 22:22

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


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