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?
I can't speak for the "Plank distance" but for the "galaxy clusters" use either the Box or Hausdorff dimension requiring that the "boxes" actually contain stars (or galaxies) in order to be counted. As you reduce the size, large numbers of the "boxes" will drop out. The dimension will depend upon what percentage of boxes (as a percentage of the total possible number of boxes) are not counted in the limit.
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