Can someone please explain the intuition behind the Minkowski dimension and measure?
I am using the definition of the Minkowski measure as $$M^{\alpha}(K)=\lim_{\epsilon\rightarrow0}\frac{\mu(V_{\epsilon}(K))}{\epsilon^{n-\alpha}} $$ Where $K\subset\mathbb{R^n}$, $\alpha\in(0,\infty)$, and $\mu$ is the $n$-dimensional Lebesgue measure and $V_{\epsilon}$ is an $\epsilon$-neighborhood of $K$.
And the definition of the Minkowski dimension is $$\dim_{M}(K)=\lim_{\epsilon\rightarrow0}\frac{N_{\epsilon}(K)}{\ln(\frac1{\epsilon})}$$ Where $N_{\epsilon}(K)$ is the minimum number of non-overlapping $n$-dimensional boxes of side length $\epsilon$ necessary to cover $K$.
I know these are largely used when working with fractals, but how does the notion of counting boxes relate.