Apparently, the closest thing I've found would be normal number http://mathworld.wolfram.com/NormalNumber.html

But requiring that every finite words occurs is weaker than this property. So I'm wondering if there are any study on this topic.

My original goal is to find a criterion for a Büchi automaton not to recognize some infinite word like this. There's been a post here Proof that the $\omega$-language consisting of all words containing every finite word as a factor is not rational/regular , but there were no references to such a class of words.

Maybe all this is obvious to specialists, but I couldn't find anything with our universal friend google :)

Thank you!

  • 4
    $\begingroup$ I've used the term "universal" for this. $\endgroup$ – Joel David Hamkins May 18 '14 at 10:09
  • $\begingroup$ In dynamical systems, this property is called transitive. $\endgroup$ – Anthony Quas May 19 '14 at 3:04

One term that is used is disjunctive sequence. The linked article mentions some references, including an overview (from 1997) by Calude, Priese, and Staiger.

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