Fourth Homework Set Answers

4. (a) The gasket has dimension log(3)/log(2), the Cantor set has dimension log(2)/log(3). Then the union of the gasket and Cantor set has dimension max{log(3)/log(2),log(2)/log(3)} = log(3)/log(2) = 1.58. The product of two Cantor sets has dimension 2log(2)/log(3) = log(4)/log(3) = 1.26. The union has a higher dimension than the product.

(b) The typical intersection of sets A and B in a space of dimension E is given by dim(A) + dim(B) - E.

For E = 2, the dimension of a typical intersection is 1.17.

For E = 3, the dimension of a typical intersection is 0.17.

For E = 4, the dimension of a typical intersection is -0.83. So typically two gaskets do not intersect in 4-dimensional space.

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