Fourth Homework Set Answers

6. (a) The dimension of the union is the max of the dimensions, so dim(A ∪ C) = 1 implies dim(C) = 1. A product of Cantor sets made of N = 4 pieces, each scaled by a factor of r = 1/4, suffices.

(b) There is no set D: the intersection of A and D is a subset of A, and so the dimension of the intersection cannot exceed the dimension of A, itself less than 1.

(c) The dimension of the product is the sum of the dimensions, so we must have 1 = dim(A) + dim(B). Because dim(A) = log(2)/log(3), we see

dim(B) = 1 - log(2)/log(3) = (log(3) - log(2))/log(3) = log(3/2)/log(3).

Unfortunately, we cannot make a fractal of 3/2 pieces, each scaled by a factor of 1/3.

We find a number x so the numerator xlog(3/2) is the log of some integer, say log(2). That is, log(2) = log((3/2)x). This gives

2 = (3/2)x

Taking logs and solving for x gives x = log(2)/log(3/2). Then the dimension of B is

xlog(3/2)/(xlog(3)) = log(2)/log(3x).

That is, the fractal B is made of N = 2 pieces, each scaled by a factor of r = (1/3)x, about 0.153.

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