let $J=S \cap D $,$G=S \cup D$,sort $G$,$a\_n \in G$.


Function $\gamma (n,s)=\frac{\Sigma\_{a\_i \in J}^n a\_i^s}{\Sigma\_{i=1}^n a\_i^s}$.


Given S ,a non computably enumerable set,is there a computably enumerable set D Such that $\lim\_{n \to \infty}\gamma (\infty,0)=1$? Or under what condition $\lim\_{n \to \infty}\gamma (\infty,0)=\frac {1}{2}$,and so on?


If such limit does not exist,is it bounded? $L \leq \gamma (n,0) \leq M $?


Especially when S is productive set or immune set,can we approximate S by such a approach?
 Are there any results about such questions?


A logician expresses the question as how about $\inf \lim\_n\frac{ |\{m\leq n \mid m\in S \cap D\}|} {|\{m\leq n\mid m \in S \cup D\}|}$ ?



How to approximate non-computably recursive set by computably recursive set 
If $S$ is selected randomly under the fair-coin (Lebesgue-Cantor) measure on $2^\omega$ then $S$ will be immune and $S$ will asymptotically contain $1/2$ of the elements of $D$, plus $1/2$ of the elements of $\omega\backslash D$. To get this to be true for all recursively enumerable $D$ it is enough that $S$ is 2-random (Martin-Löf random relative to $0'$). For background you can look at the recent books Downey and Hirschfeldt, *Algorithmic randomness and complexity*, and Nies, *Computability and randomness*.


