The n-Category Café

Thanks!

Here’s a big class of examples where magnitude dimension and box dimension differ. Let $A$ be a compact subset of $n$-dimensional Euclidean space with positive Lebesgue measure, and write $A^\alpha$ for $A$ equipped with the metric $d(x,y) = \|x - y \|^\alpha$, where $\|\cdot\|$ is the Euclidean norm and $\alpha \in (0,1]$. Then $A^\alpha$ has magnitude dimension $n$ for each $\alpha$ (by the last two theorems in this paper, which generalize results of Tom’s for the case $\alpha = 1$; there are also $\ell_p$ versions of this).

On the other hand, it’s an easy exercise to check that the box dimension of $A^\alpha$ is $1/\alpha$ times the box dimension of $A^1$, thus $n/\alpha$.

Is that weird enough for you? The set $A$ can be as tame as you like, but the metric on $A^\alpha$ isn’t so easy to visualize — distances get stretched a lot at small scales and shrunk at large scales. It’s a nice related puzzle to explicitly isometrically embed an interval, equipped with the square root of the usual metric, into a Hilbert space — but it doesn’t embed into a finite dimensional subspace.

More recently, inspired by work I learned about in Banff, I proved that (under some hypotheses which are always satisfied in the cases discussed above) box dimension is less than or equal to magnitude dimension. Since Hausdorff dimension is less than or equal to box dimension, this is an improvement of a result mentioned on the slides.

This was somewhat distressing, though, since it contradicts the observation I made in the comment above. Tom helped me find the error behind the above observation. The upshot is that what I actually proved in my paper is that the magnitude dimension of such spaces $A^\alpha$ is $n/\alpha$ — precisely equal to the box dimension.

So as far as I know right now, magnitude dimension may always be equal to box dimension. This would be a big piece of support for the point of view that magnitude measures the “number of points you can see if you squint your eyes”.