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In connection to this question: Expected values of Kolmogorov complexity in a random sample

Let $n$ be number of bits. Let $A = \{0,1,2,\dots,2^{n}-1\}$ be indexed by the $n$-bits. Let $ \delta > 0$ be fixed. Is there an estimate of what fraction of integers in $A$ can have Kolmogorov complexity be greater than $\delta n$?

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    $\begingroup$ I think this is off-topic for cstheory as not research-level. However, you can flag it for migration to CS.SE. $\endgroup$
    – Artem Kaznatcheev ♦
    Jun 11, 2012 at 12:28
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    $\begingroup$ It may not be research-level to some, but I didn't know the answer, and could have imagined asking it. It might fall under the category of "easy question in subarea X not known to researchers in subarea Y" --- or I'm just illiterate :) $\endgroup$
    – Suresh Venkat
    Jun 11, 2012 at 15:16

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The argument is the same, there are only $1 + 2 + 4 + ... + 2^{\delta n - 1}$ programs shorter than $\delta n$ so there are at least $2^n - (2^{\delta n} - 1)$ integers with $K(x) \geq \delta n$.

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    $\begingroup$ Just to add, this is an example of the general phenomenon that most objects are hard. $\endgroup$
    – Tsuyoshi Ito
    Jun 12, 2012 at 1:44

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