A notion of a 'coarse', parametrized dimension of an object, where the parameter determines how finely we can distinguish (say) a very thin rod from a line

Question

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.

Answer 1

A few observations first:

1. I assume in your motivating example that the coarseness parameter $\epsilon$ is smaller than $1$, otherwise you can't even tell that cylinder apart from a point.
2. Anton's comment is exactly along the right lines. For those who have not seen it before:

Definition: a metric space $(M,d)$ has macroscopic dimension $n$ if $n$ is the smallest integer for which there exists a triple $(K,f,c)$ consisting of an $n$-dimensional simplicial complex $K$, a continuous map $f: M \to |K|$ and a constant $c > 0~$ with $\text{diam}(f^{-1}(y)) < c$ for all $y \in K$.

In the context of the current question, you could use the following related -- but much less general -- idea. Let $A$ be the subset of interest in a normed linear space $V$. Just define $d_\epsilon(A)$ to be the smallest $n$ for which there exists an $n$-dimensional simplicial complex $K$ and a continuous injection $i: |K| \hookrightarrow V$ so that $A$ is contained in the $\epsilon$-dilation of $i(|K|)$. The $\epsilon$-dilation of any subset $B$ of $V$ is defined to be the union of $\epsilon$-balls around points in $B$.

In general it will be hard to know which simplicial complexes will achieve the minumum without already knowing the answer: but at least one can always get upper bounds.

Answer 2

Something like $f(r)=\log_2 N_A(r)/\log_2 N_A(r/2)$, where $N_A(r)$ denotes the smallest number of $r$-balls that covers $A$ might work.

If you had a wire $W$ of cross-sectional radius $\delta$ and length 1, then for $r\gg\delta$, $N_W(r)\approx 1/\delta$, while for $r\ll\delta$, $N_W(r)\approx \delta^2/r^3$. When you compute the $f$, you get roughly what you need.

This is basically $$ \frac{dN_A(r)}{d\log r}. $$

The only difficulty is that $N_A$ isn't a differentiable function.