The theoretical definition for Hausdorff, box-counting, and so forth dimensions is clear, but I do not see how this is applied to physical measurements. For example, I have read the following two claims:

(a) that the dimension of space-time falls from 4 down towards zero as one gets in the neighbourhood of the Planck distance

(b) that certain galaxy clusters have a fractal dimension.

Whether or not these two claims are valid is not the point here. The question is: how can one arrive to such conclusions? What are they scaling?