I recently learned of the concept of box dimension (also called the Minkowski-Bouligand dimension) as a way of measuring the dimension of a set (with use in describing the non-integer dimensions of some fractals) - it's sometimes compared to the Hausdorff dimension and in some cases can be simpler to calculate. The box-dimension of a set $S$ contained in $d$-dimensional Euclidean space considers the occupancy of a grid of equal size boxes of lateral dimension $\epsilon$. In particular, calling the number of boxes containing at least one element of $S$ $N(\epsilon)$, we have $$B(S) = \lim_{\epsilon \to0} \frac{\log(N(\epsilon))}{\log(\frac{1}{\epsilon})}$$
For example, the box dimension of the set $S={1, 1/2, 1/3,..}$ on the real line (i.e. $\frac{1}{n}$ for integers $n>0$) is $1/2$, as noted in this question.
What is the box dimension of $\cup_{m=1}^\infty \{ \frac{1}{n^m}\mid n=1,2,3,\ldots \}$?
Here are my thoughts. I believe that in general, $\{ \frac{1}{n^m}\mid n=1,2,3,\ldots \}$ has box dimension $\frac{1}{m+1}$ for positive $m$, and this was shown by Sarvesh in a separate question of mine. I believe the box dimension of a finite union of sets is the maximum of the box dimensions of those sets. However, this is not the case for an infinite union. Nevertheless, it might be tempting that the box dimension could turn out to be $1/2$ once again; in any case, the box dimension is between $1/2$ and $1$ inclusive.
