I apologize for the clumsy wording of the title-- what I'm looking for is a notion of an integer-valued dimension $d_{\epsilon}$, which we parametrize by a real positive number $\epsilon$, of, say, a connected subset $A$ of your favorite normed linear space.

The number $\epsilon$ is our 'coarseness' parameter, for which I have the following example in mind: consider a cylinder with unit length and a given radius inside $\mathbb{R}^3$. Then if $\epsilon$ is too big relative to the radius of the cylinder, then $d_{\epsilon}(A)=1$ since at this level of coarseness, the dimension function fails to distinguish between such an object and a line segment in $\mathbb{R}^3$. On the other hand, at a 'fine enough' (small enough) $\epsilon$, we have $d_{\epsilon}(A)=3$ since we can distinguish the cylinder structure. Ideally, we have a degeneration in general such that $d_0$ of any set is just the 'usual' (say Euclidean) dimension of the set.

Are there classical notions of dimensionality for such 'thin' structures that are effectively of a smaller dimension if you 'look far enough away'? The closest I come to this notion would be some sort of capacity dimension that helps distinguish how close certain fractals are to being 'line-like' versus 'area-like' (in the case of fractional dimensions between 1 and 2). But the idea I'm interested in is perhaps simpler than this.