I am self-studying introduction to ergodic theory and found this problem (application of Poincare recurrence theorem) which I couldn't do.
I need to show the set, $E = \{x \in [0,1]: \text{decimal representation of $x$ has $5$ infinitely often}\}$, has measure 1. How can this be extended to show that almost all elements in $[0,1]$ contain the block (this sequence of numbers appear consecutively and in the same order) $123$ in decimal representation infinitely many times?
What I did: I know, by Poincare recurrence theorem, for almost all elements in $[0.5,0.6)$ have 5 infinitely often in the decimal representation. I am not sure how the Poincare recurrence theorem can be used to extend this to the whole of $[0,1]$. Any hints would be helpful.
You need to show that almost all points ultimately fall in your interval $[0.5,0.6)$ when you shift the decimal representation. What are the points who do not fall in that interval? The ones with no 5 in the decimal expansion. Can you show that the set of these points is of zero Lebesgue measure?
Then a small computation is needed to conclude. Let $I = [0.5,0.6)$, $B$ the set of points in $I$ coming back infinitely many times in $I$ and $A$ the set of points entering $I$ at some point, $T$ the shift of the decimal point, aka multiplication by 10 mod 1. Then $$ \mu(I \backslash B) = 0, \quad \mu(T^{-n}I \backslash B) = \mu(T^{-n}(I \backslash B)) = 0, \quad \mu(\cup T^{-n}I \backslash B) =0 $$ and you should be able to conclude.