This means we should report the fractional dimension of an object not just as a single number, but use a continuous function that takes in a scalar describing the scale level, and telling us what the fractional dimension at that particular scale is.
Also potentially relevant is the magnitude function, which is a function |tA| of a space A and a real-valued scale factor t, and asymptotically grows as O(tdimA) where dimA is A’s Minkowski dimension (which usually agrees with Hausdorff dimension).
The relevant keyword is covering number.
Also potentially relevant is the magnitude function, which is a function |tA| of a space A and a real-valued scale factor t, and asymptotically grows as O(tdimA) where dimA is A’s Minkowski dimension (which usually agrees with Hausdorff dimension).
Thanks. The function I am describing can be derived from the covering number function, but is also distinct.