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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.

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Are you sure you want it to be integer-valued? While you don’t mention continuity, this seems like another desirable property (from the intuition you sketch); it can’t be simultaneously integer-valued, non-constant, and continuous in $\varepsilon$, but of these three, “integer-valued” seems like the one one might most reasonably abandon. So in your example, for instance, it might not be equal to 1 for any $\varepsilon > 0$, but it would approach 1 as $\epsilon$ goes to 0. – Peter LeFanu Lumsdaine May 27 '13 at 2:38
Another way to preserve some integrality but also to have some continuity is if instead of a single number, you allow a probability distribution. In physics, it makes sense to take some measurements at length scale $\epsilon$, and have some probability that the measured dimension is $1$, and some probability that the measured dimension is $3$. – Theo Johnson-Freyd May 27 '13 at 3:20
Macroscopic dimension is exactly the one you decribing. – Anton Petrunin May 27 '13 at 4:37
You should take a look at persistent homology, where you can get homology groups that depend on a coarseness parameter $\epsilon$. A good starting point: – Bryan Clair May 27 '13 at 6:35
Thank you all for your helpful comments. I'm not too concerned about continuity everywhere (at least as far as the domain of ϵ is concerned), since I think it might actually be more natural to preserve jumps as we suddenly resolve extra structure, at least for simple objects in my particular application. But it is an important point to consider. @Theo, would you happen to know of a source where I can take a look at your idea of 'dimension up to measurement precision'? That seems like a very useful concept. – ithmath May 28 '13 at 1:36
up vote 0 down vote accepted

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.

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Thanks very much for your comment, Vidit-- it eases my concern that there is a thread I can follow to get a more rigorous notion of what I need. But just to get a handle on the definition you presented, my very naïve intuition on macroscopic dimension is that if we can contract a certain space onto a clever choice of $n$-dimensional simplex such that the contraction does not move things too much (greater than $c$), then we can assert that the macroscopic dimension is no greater than $n$. Is this more or less the case? – ithmath May 28 '13 at 1:42
ithmath, that's about right. Of course, the map $f: M \to |K|$ from the definition need not be a "contraction" in the strict sense (of strong deformation retraction): it is just any continuous map with control on sizes of point inverses. And yes, the fact that you can construct a triple $(K,f,\epsilon)$ for some choice of $n$ only tells you that the macroscopic dimension is $\leq n$ at your scale $\epsilon$. – Vidit Nanda May 28 '13 at 3:22

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.

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