Is it possible that part of sequence is more complex than all sequence because the best way to encode it is to use the complete sequence and starting and ending positions of the fragment. Maybe, for example, string of hexadecimal digits of $\pi$. Or something else.

Define $f(n) = \lfloor 2^n \cdot \Omega \rfloor$, that is, $f(n)$ is the first $n$ bits of Chaitin’s constant interpreted as a number written in binary. I am trying to figure out if $f(n)$ can have infinitely many prime values. To show that it cannot, it would suffice to find a way of compressing these […]

I’m looking for a proof, that $$ \sum_{i=0}^{\lambda n} \binom{n}{i} \le 2^{nH(\lambda)} $$ with $n>0$, $0 \le \lambda \le 1/2$ and $ H(\lambda)=-[\lambda log \lambda + (1-\lambda) log (1-\lambda)] $. Context: This shows, that if there is a bitstring with a ratio of ones and zeros (‘pick i from n’ where i is smaller than […]

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