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Let $I= (3,4)$, and write the decimal expansion of $x\in I$ in the form $x= 3.d_1d_2d_3 . . .$ such that $d_k$ is not eventually 9,
(this makes the decimal representation of $x$ unique).
Let $c_k$ be the decimal digits of $\pi$, so that $\pi = 3.c_1c_2c_3 . . .$

  • What is the measure of the set $A\subset I$ of points $x$ that have infinitely many decimal digits in common with $\pi$ ?
    ($x\in A$ if and only if $d_k= c_k$ for infinitely many k).
  • What is the measure of the set $B\subset I$ of points $x$ that have infinitely many blocks $d_kd_{k+1} . . . d_{2k−1}$ in common with $\pi$ ?
    ($x\in B$ if and only if $d_k = c_k,\ d_{k+1} = c_{k+1},\ . . . , d_{2k−1}= c_{2k−1}$ for infinitely many blocks).
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  • $\begingroup$ This problem has been solved! We take the complement of the set and find its measure to be 1. $\endgroup$
    – NotaChoice
    Dec 25, 2020 at 22:51

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