Some Algebra of Dimensions

Dimensions of products

The Cantor set and line segment example suggests a plausible relation is

db(A × B) = db(A) + db(B)

Here A × B denotes the Cartesian product of A and B. Most simply,

A × B = {(a, b): a belongs to A and b belongs to B}

For example, if A is a Cantor set that lies in a line, and B is a line segment that lies in a line, then A × B lies in a plane, a line in one direction and a line in another direction.

Here is a sketch of the proof in a simple case.

Suppose NA(r) is the number of boxes (line segments in this example) needed to cover A, and NB(r) is the number of boxes (line segments again) needed to cover B.

To cover A × B, form boxes made from the boxes covering A and the boxes covering B. In the example, we form squares from the product of two line segments, one in the x-direction and one in the y-direction.

Then NA × B(r) = NA(r)⋅NB(r) and so

Log(NA × B(r)) = Log(NA(r)⋅NB(r)) = Log(NA(r)) + Log(NB(r))

Consequently,

Log(NA × B(r))/Log(1/r) = Log(NA(r))/Log(1/r) + Log(NB(r))/Log(1/r)

Taking the limit as r → 0 gives

db(A × B) = db(A) + db(B)

We have made simplifying assumptions here, for example, that the same scaling factor r works for both A and B.

In the most general case, the best result is

dim(A × B) ≥ dim(A) + dim(B)

Here dim refers to the Hausdorff dimension, more mathematically demanding but equal to ds for exactly self-similar shapes. See chapter 7 of Falconer.

The addition formula does hold for regular Cantor sets (for example, consisting of N = 2 pieces scaled by a factor of r < 1/2). So we can compute the dimension of some self-affine sets.

For example, the product of Cantor sets of dimension Log(2)/Log(3) and Log(2)/Log(4) has dimension Log(2)/Log(3) + Log(2)/Log(4).

Return to the algebra of dimensions.