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 ∈ [ 0 , 1 ] : 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.

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I am self-studying introduction to ergodic theory and found this problem (application of Poincare recurrence theorem) which I couldn&#039;t do.I need to show the set,   E  =  {  x  ∈  [  0  ,  1  ]  :      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.

kuncwadi17 · Accepted Answer

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  μ  (  I  ∖  B  )  =  0  ,    μ  (      T          −      n        I  ∖  B  )  =  μ  (      T          −      n        (  I  ∖  B  )  )  =  0  ,    μ  (  ∪      T          −      n        I  ∖  B  )  =  0and you should be able to conclude.

I am self-studying introduction to ergodic theory and found this problem (application of Poincare re

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