The question is very direct.
How can I calculate the dimension of capacity of the set: $\mathbb{Q}\cap[0,1]$?
I know that the dimension of capacity of a set $A\subseteq\mathbb R^n$ ( but I' m not $100\%$ sure if I could define the same for$\mathbb{Q}^n$) is $\dim A=\displaystyle\lim_{\varepsilon\to 0}\frac{inN(\varepsilon)}{\ln\left(\frac{1}{\varepsilon}\right)}$ if the limit exists.
Now I find that for a closed set the dimensions of Minkowski-Bouligand and Hausdorf-Besicovitch are the same $D_{0}=D_{MB}=D_ {HB}$ while if a set is not closed, the Hausdorff-Besicovitch dimension may differ from the other two. Now I don't really know if this can help me and if I can use it in some way...
Edit:
The fact is that the box-counting doesn't distinguish dense subset. For this reason I've tried to find some other methods to resolve this problem and I find something about the two other type of dimension even if I don't really know if these can help me. –