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.