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Let $\mathbb{N}$ denote the set of non-negative integers. We can identify every bitstream, i.e. a function $s:\mathbb{N}\to \{0,1\}$, with some $A\in{\cal P}(\mathbb{N})$: take $A = s^{-1}(\{1\})$.

Given any $S\subseteq \mathbb{N}$ we define maps $\mu_S^+, \mu_S^-:{\cal P}(\mathbb{N})\to[0,1]$ by setting, for every $A\in{\cal P}(\mathbb{N})$, $$\mu^{+}_S(A)= \lim \sup_{n\to\infty}\frac{|A\cap S \cap\{1,\ldots,n\}|}{1+|S\cap \{1,\ldots,n\}|}, \text{ and } \mu^{-}_S(A)= \lim \inf_{n\to\infty}\frac{|A\cap S \cap\{1,\ldots,n\}|}{1+|S\cap \{1,\ldots,n\}|}.$$

We say that $A$ is well-balanced with respect to $S$ if $\mu^+_S(A) = \mu^-_S(A) = 1/2$.

We say that $A\subseteq \mathbb{N}$ is computationally random if for every computable set $S\subseteq\mathbb{N}$, the set $A$ is well-balanced with respect to $S$. Morevover, we say that the bitstream $s:\mathbb{N}\to\{0,1\}$ is computationally random if $s^{-1}(\{1\})$ is computationally random.

Note that neither the Thue-Morse sequence or the Champernowne sequence $C_2$ are computationally random.

Question. Is every computationally random bitstream normal (that is, every finite $01$-string appears infinitely often)?

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  • $\begingroup$ The linked article has a variety of definitions for algorithmic randomness, focusing on Martin-Lof randomness. Which one of those do you think might be identical with yours? $\endgroup$
    – Matt F.
    Jul 1, 2019 at 10:36
  • $\begingroup$ Your randomness notion is Church stochasticity, which is distinct from each of the randomness notions on the wikipedia page. $\endgroup$ Jul 1, 2019 at 11:08
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    $\begingroup$ Thanks @DanTuretsky , I will check out Church stochasticity. I will change the question to make it more precise. $\endgroup$ Jul 1, 2019 at 12:29
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    $\begingroup$ You need to be more clear on your definition of "normal". I think you mean to say normal in base 2, meaning that for every length $l$, every finite binary string of length $l$ appears with asymptotic density $\frac{1}{2^l}$, not just infinitely often. $\endgroup$
    – James
    Jul 1, 2019 at 14:07
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    $\begingroup$ You want to restrict $S$ to be infinite. It's obviously not true if $S = A \cap \{1,\ldots,N\}$ for some $N$. $\endgroup$ Jul 1, 2019 at 14:47

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Let $s$ be a computationally random bitstring. Consider $\tilde{s}$ defined by $\tilde{s}(2n) = \tilde{s}(2n+1) = s(n)$. Then $\tilde{s}$ should also be computationally random, but it does not contain any $010$ or $101$.

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  • $\begingroup$ Very slick argument -- thanks! $\endgroup$ Jul 2, 2019 at 18:31

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