The problem I'm trying to tackle is to show that for a Kolmogorov-random number of length $n$, the amount of $1$'s in its binary representation is greater than $n/4$.
My only idea so far, is to use a Turing machine that takes indices of $1$'s as an argument and produces a binary sequence accordingly. Then, for instance, we could say the amount of ones is at least $\sqrt n$ as (roughly) $C_U(n)\leq \sqrt n\log n < n$ for $n$ large enough (each index is at most of length $\log n$). But this doesn't allow us to infer there are more than $n/4$ ones.
Is there any other way to generate some given number that would prove the statement in question?