For any positive integer $n$, let $X\_n$ be the family of all subsets of $\{1,2,\cdots,n\}$. 


Let $(X\_n,d)$ be the metric space such that
$$d(A,B)=|\,A\triangle B\,|,\ \forall A,B\in X\_n$$
where $A\triangle B$ is the symmetric difference of $A$ and $B$.


Let real $k>0$ (be fixed), and let $\ S\_n\subseteq X\_n\ $ satisfy:


      for every $F\in X\_n$, there exists $E\in S\_n$ such that $d(E,F)<k\sqrt{n}$.


Is


$$ \liminf\_{n\rightarrow\infty}\frac{\mid S\_n\mid}{\mid X\_n\mid}>0\,? $$



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Now I know the answer is no! I want to know more about quotients
$\ \frac{|S\_n|}{2^n}\ $ as $\ n\rightarrow\infty\, ?$



A question about a $2^n$-point metric space
No: let $S\_n$ be the collection of subsets of $\{1,\ldots,n\}$ whose size is a multiple of $\lfloor k\sqrt n\rfloor$. Then every subset of $\{1,\ldots,n\}$ may be approximated by an element of $S\_n$ with error at most $\frac 12\lfloor k\sqrt n\rfloor$. The ratio $|S\_n|/|X\_n|$ is approximately $1/(k\sqrt n)$.


