Is There a Reason One Should Try To Calculate Hausdorff Or Minkowki Dimensions?

Two examples off the top of my head:

- The different fractal dimensions are invariants under certain classes of functions. For example, if I know two sets have different Hausdorff or Minkowski dimensions, then there cannot be a bi-Lipschitz map between them; in particular, if they are both compact, then they are not diffeomorphic to each other. (I'm a bit rusty on this so correct me if I'm mistaken.)
- The Marstrand projection theorem tells us about the shadows of fractal sets. For example, if S is a compact subset of R^n of Hausdorff dimension 1.5, then its projection onto a random line will almost surely have positive length, while its projection onto a random 2-dimensional plane will almost surely have zero area.
A lot of Aaron Naber’s work on singular solutions to nonlinear geometric PDEs (Yang-Mills and the like) seeks to find the largest dimension of their stratified singular sets. Most of the big theorems involve showing something like the singular sets have Hausdorff dimension n-4 where n is the dimension of the domain of definition.

In general though, geometric measure theory heavily uses Hausdorff and Minkowski dimension as a measure of size of certain sets of interest.
In physics, some of these fractal dimensions (in particular self-avoiding random walks, and look up "percolation" for a lot of these) tell you about how some things (like free energy, correlation length, etc) scale as you get near the critical point of phase transitions.
To add to what has already been mentioned: the Minkowski dimension of a set is intimately related to geometric and spectral properties of that set.

For example, it shows up in the leading order asymptotics for the tube formula of that set (c.f. how big is a neighborhood of the set?) and for the Dirichlet problem eigenvalue counting function (c.f. what are the frequencies of a drum with that set as a boundary?) among other things.

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