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Let $f:\mathbb{N} \to \{0,1\}$ be the Ehrenfeucht-Mycielski sequence. The first few digits of the sequence are: $$010011010111000100001111\ldots$$

For any $k\in\mathbb{N}$ let $s(k) = \sum_{i=0}^k f(i)$. It is conjectured that $\lim_{n\to\infty} s(n)/n = 1/2$.

Question. Is it known that $\liminf_{n\to\infty} s(n)/n = \limsup_{n\to\infty} s(n)/n$?

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    $\begingroup$ OEIS has several entries related to the Ehrenfeucht-Mycielski sequence, see oeis.org/… – the main entry is oeis.org/A038219 which links to further work on the sequence. Perhaps some of the links give some information on the limit questions. $\endgroup$
    – Gerry Myerson
    Commented Sep 28 at 0:59
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    $\begingroup$ Quoting the Wikipedia page you have linked, with your notation: "it is known that every limit point of the sequence of values $s(n)/n$ lies between 1/4 and 3/4 inclusive." $\endgroup$
    – domotorp
    Commented Sep 28 at 4:48
  • $\begingroup$ Have you had a look at the links at the oeis page, Dominic? $\endgroup$
    – Gerry Myerson
    Commented Sep 30 at 10:39
  • $\begingroup$ Thanks Jerry - but I haven't found a hint as to whether the limit exists (aka lim inf equals lim sup) $\endgroup$
    – Dominic van der Zypen
    Commented Oct 1 at 14:22

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