For subsets of $\mathbb{R}^n$, I want a notion of dimension $\operatorname{dim}$ verifying:

If $\operatorname{dim}(A) = d$, then there's a constant $C$, depending only on $A$, $d$ and $n$, such that for every $\varepsilon >0$ it's true that $A$ can be covered by at most $C\big ( \frac{1}{\varepsilon}\big )^d$ balls of radii less than $\varepsilon$.

I think that the upper Minkowski dimension could work, but I'm not sure. Minkowski dimensions are defined for every subset $A$ of $\mathbb{R}^n$ by $$ \overline{\operatorname{dim}}_M (A)= \inf{\{ s:\limsup_{\varepsilon\downarrow0}{N(A,\varepsilon)\varepsilon^s = 0}\}}$$ and $$ \underline{\operatorname{dim}}_M (A)= \inf{\{ s:\liminf_{\varepsilon\downarrow0}{N(A,\varepsilon)\varepsilon^s = 0}\}},$$ where $$N(A,\varepsilon) = \min{\{ k : A \subset \bigcup_{i=1}^{k}{B(x_i,\varepsilon)} \quad \text{for some}\; x_i \in \mathbb{R}^n\}}.$$ We call $\overline{\operatorname{dim}}_M (A)$ and $\underline{\operatorname{dim}}_M (A)$ the upper and lower Minkowsi dimension respectively.

If $\overline{\operatorname{dim}}_M (A) = d$, then for any $\delta >0$ there is a constant $C$, depending on $\delta$ and $A$, such that for any $\varepsilon >0$ the set $A$ can be covered by at most $C\big ( \frac{1}{\varepsilon}\big )^{d+\delta}$ balls of radii less than or equal to $\varepsilon$. This is similar to, but not exactly what I needed. Nevertheless, this could be fixed if the infima in the defininitions of Minkowski dimensions were minima under reasonable hypothesis.

So, my questions are:

When do the infima are minima?

If the answer is not "always", what are some counterexamples?

Is there some other notion of dimension that can work?