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 a non 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 (\infty,0) \leq M $?

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

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